Week 1

Section 2.1 Week 1

Subsection Monday

Subsubsection Welcome to the Generous Arena

sites.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want%20-%20final

(2019), Penelope Maddy argues that set theory forms a Generous Arena for mathematics:

a unified setting with a Shared Standard of Proof, and a Meta-mathematical Corral so that formal techniques can be applied to all of mathematics at once.
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As interesting and alive as they are, we won’t pursue questions of foundations in this course. But I want to argue that topology is a Generous Arena for all — or at least nearly all — things geometric in mathematics. Given a mere whiff of geometry, some sense of closeness or continuity or paths, topology is lurking and the experience and labor of generations of mathematicians has revealed that the formalism of topology offers a common language in which these notions can be captured and made precise.

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This is somewhat at odds with the common conception of topology as "rubber sheet geometry" — a coffee cup is the same as a donut! the Möbius strip only has one side! Euler’s formula for planar graphs! Khovanov homology distinguishes the unknot! These are fine, even wonderful, topics and results, and we will discuss some of them in due course. But they do poor service in capturing the stunning breadth and extreme utility of topology in contemporary mathematics.

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To make this claim concrete, here is an incomplete list of settings in which topological ideas (spaces, continuity, compactness, connectedness, homotopy, homology, and their descendants) function as a shared language:

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Manifolds and singular spaces. Smooth manifolds, manifolds with boundary, orbifolds, and stratified spaces; the passage from local geometry to global invariants; and the basic fact that many geometric problems are really problems about how local pieces glue.
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Configuration and moduli spaces. Spaces of points in a manifold, spaces of embeddings, and moduli of geometric structures; topology keeps track of how parameters vary continuously, and it detects global features that no coordinate chart can see.
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Algebraic geometry and its avatars. Schemes as locally ringed topological spaces, complex varieties viewed as topological spaces, étale/profinite shadows of spaces, and the way cohomology packages intersection data and subtle “hidden” structure.
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Category theory as geometry. Categories as spaces (via nerves and classifying spaces), higher categories as higher-dimensional geometry, and the guiding principle that “equivalence” is often the right notion of sameness.
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Dynamics and flows. Long-term behavior of iterated processes and differential equations; attractors, recurrence, and qualitative features that persist under perturbation.
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Combinatorics with geometry inside. Graphs, posets, and simplicial complexes; topological invariants that constrain what combinatorial objects can exist and how they must behave.
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Topological data analysis. Turning point clouds into simplicial complexes and extracting stable, noise-robust features; using homology as a kind of geometric summary statistic.
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Physics and symmetry. Spaces of fields, gauge symmetry, and phase spaces; invariants arising from quantization and from the topology of symmetry actions.
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Computation and invariants. Algorithms that compute topological signatures; the practical question of which invariants are computable and what they reveal (and conceal) about geometric input.
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Arithmetic geometry. Number theory has its own natural topologies: the profinite Galois group \(\mathrm{Gal}(\overline{K}/K)\) assembled from finite quotients, the \(p\)-adic integers \(\mathbb{Z}_p\text{,}\) and the adeles \(\mathbb{A}_K\text{,}\) where “closeness” is measured \(p\)-adically and via local-global structure.
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Our aim in this course is to build enough of the basic toolkit to recognize these phenomena when they appear, and to compute invariants that remain meaningful when coordinates, metrics, and other rigid structures are unavailable (or irrelevant).

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Subsubsection So what is a topology?

Whatever a topology is, it is should specify a notion of convergent sequences. That is, if \(X\) is a topological space (to be defined in due course) and \(x\colon \mathbb{N}\to X\) is a sequence in \(X\text{,}\) then the statement "the limit of \(x\) is \(L\)" should make sense.

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Now if \(X = \mathbb{R}^n\) is Euclidean space, we know what this means:

for all \(\varepsilon>0\) there exists \(N\in\mathbb{N}\) such that \(n>N\) implies \(d(x_n,L) \lt \varepsilon\)
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where \(d\) is the standard distance function on \(\mathbb R^n\text{.}\) If we focus on the role of \(d\text{,}\) we’ll arrive at the notion of a metric space, which is not where we’re headed presently. Instead, write \(B(L,\varepsilon)\) for the ball of points in \(\mathbb R^n\) distance less than \(\varepsilon\) from \(L\text{.}\) We can then recast convergence of \(x\) to \(L\) as

for all \(\varepsilon>0\) there exists \(N\in\mathbb{N}\) such that \(n>N\) implies \(x_n\in B(L,\varepsilon)\)
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for every ball \(B\) centered at \(L\) there exists \(N\in\mathbb N\) such that \(n\gt N\) implies \(x_n\in B\text{.}\)
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This suggests that a topology on \(X\) might be specified by a basis \(\mathscr B\) of subsets of \(X\) with similar properties to those of open balls in \(\mathbb R^n\text{.}\)

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Definition 2.1.1.

A (topological) basis on a set \(X\) is a collection of subsets \(\mathscr B\subseteq 2^X\) satisfying the following two properties:

Covering: 🔗: \(\bigcup_{B\in \mathscr B} B = X\text{;}\) in other words, every \(x\in X\) is contained in some \(B\in\mathscr B\text{.}\)
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Filtered: 🔗: for all \(B_1,B_2\in\mathscr B\) and all \(x\in B_1\cap B_2\) there exists \(B_3\in\mathscr B\) such that \(x\in B_3\subseteq B_1\cap B_2\text{.}\)
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Checkpoint 2.1.2.

Verify that the collection of open balls in \(\mathbb R^n\) is a basis.

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Definition 2.1.3. Convergence with respect to a basis.

Suppose \(X\) is a set equipped with a basis \(\mathscr B\subseteq 2^X\) and that \(x\) is a sequence in \(X\text{.}\) Then \(x\) converges to \(L\in X\) with respect to \(\mathscr B\) :iff for all \(B\in \mathscr B\) containing \(L\) there exists \(N\in\mathbb N\) such that \(n\gt N\) implies \(x_n\in B\text{.}\)

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Given a basis, we can define open sets:

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Definition 2.1.4. Open sets with respect to a basis.

Suppose \(X\) is a set equipped with a basis \(\mathscr B\subseteq 2^X\text{.}\) A subset \(U\subseteq X\) is called open with respect to \(\mathscr B\) :iff \(U\) is a union of basis sets; equivalently, there exists \(S\subseteq \mathscr B\) such that \(U = \bigcup_{B\in S}B\text{.}\)

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Remark 2.1.5. Nothing is open.

The empty set \(\varnothing\subseteq X\) is automatically open: \(\varnothing = \bigcup_{B\in \varnothing}B\text{.}\) This is a computation, not a convention. Indeed, \(x\in \bigcup_{B\in S}B\) :iff there exists \(B\in S\) such that \(x\in B\text{.}\) Since there is no \(B\) in \(S=\varnothing\text{,}\) we learn that the set \(\bigcup_{B\in \varnothing}B\) has no elements!

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Remark 2.1.6. Everything is open.

Recall that \(\mathscr B\) covers \(X\text{.}\) As such, \(X = \bigcup_{B\in\mathscr B}B\) is open.

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Remark 2.1.7. Unions of opens are open.

Given open sets \(U = \bigcup_{B\in S} B\) and \(V = \bigcup_{B\in T} B\) where \(S,T\subseteq \mathscr B\text{,}\) we find that

\begin{gather*} \end{gather*}

is also open since \(S\cup T\subseteq \mathscr B\) as well.

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Checkpoint 2.1.8. Arbitrary unions of opens are open.

Extend the above reasoning to argue that arbitrary unions of open sets are open. Your task is to show that for any collection of open sets \(\mathscr U\text{,}\) the set \(\bigcup_{U\in\mathscr U}U\) is open as well. You may not assume that \(\mathscr U\) is finite; in particular, an inductive argument won’t work!

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Remark 2.1.9. Finite intersections of opens are open.

It is also the case that if \(U,V\) are open with respect to a basis \(\mathscr B\subseteq 2^X\text{,}\) then \(U\cap V\) is open with respect to \(\mathscr B\) as well. One can then argue by induction that any finite intersection of open sets is open.

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Checkpoint 2.1.10.

Verify the claims of Remark 2.1.9.

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This leads us to the notion of a topology on a set.

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Definition 2.1.11. Topology on a set.

A topology on a set \(X\) is a collection of subsets \(\tau\subseteq 2^X\) (called open sets) that is closed under arbitrary unions and finite intersections. A set \(X\) equipped with a topology \(\tau\) is called a topological space. We will often abuse notation and write \(X = (X,\tau)\) when the topology is understood from context.

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Remark 2.1.12. On a redundant axiom.

Some authors further specify that \(\varnothing\) and \(X\) are open (that is, elements of \(\tau\)). Since \(\bigcup \varnothing = \varnothing\) and \(\bigcap \varnothing = X\) (when taking intersections of subsets of \(X\)), these conditions actually follow from closure under unions and finite intersections. Nonetheless, it is certainly important that \(\varnothing\) and \(X\) are open sets in every topology on \(X\text{.}\)

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We have seen that every basis induces a topology by declaring the open sets to be unions of basis sets. It is also the case that every topology is a basis. We will now define sequence convergence with respect to a topology, and it will be your task to confirm that convergence with respect to a basis is the same thing as convergence with respect to the topology induced by that basis.

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Definition 2.1.13. Open neighborhoods.

Suppose \((X,\tau)\) is a topological space and \(x\in X\text{.}\) An open neighborhood of \(x\) (with respect to \(\tau\)) is an open set \(U\in \tau\) that contains \(x\text{.}\)

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Definition 2.1.14. Sequence convergence with respect to a topology.

Suppose \((X,\tau)\) is a topological space and \(x\) is a sequence in \(X\text{.}\) We say that \(x\) converges to \(L\in X\) :iff for every open neighborhood \(U\) of \(L\) there exists \(N\in \mathbb{N}\) such that \(n\gt N\) implies \(x_n\in U\text{.}\)

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Checkpoint 2.1.15. Sequence convergence with respect to basis versus topology.

Let \(X\) be a set, \(\mathscr B\) be a basis on \(X\text{,}\) and \(\tau\) be the topology generated by \(\mathscr B\text{.}\) Show that a sequence \(x\colon \mathbb N\to X\) converges to \(L\in X\) with respect to the basis \(\mathscr B\) if and only if it converges to \(L\in X\) with respect to the topology \(\tau\text{.}\)

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Subsubsection Examples of topologies

It is now our lot to recite some fundamental examples of topologies. Of course, you won’t really have a feel for these until you start working with topological spaces, but we have to start somewhere. The careful reader will verify that each example is indeed a topology (closed under arbitrary unions and finite intersections).

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Example 2.1.16. Discrete topology.

The discrete topology on a set \(X\) is \(\tau_{\text{disc}} := 2^X\text{.}\) That is, in the discrete topology, every subset of \(X\) is open.

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Example 2.1.17. Concrete topology.

The concrete topology on a set \(X\) is \(\tau_{\text{conc}} := \{\varnothing, X\}\text{.}\) Other authors sometimes call this the trivial, indiscrete, or codiscrete topology.

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Example 2.1.18. Sierpiński space.

The Sierpiński space is the set \(\mathbb 2 = \{0,1\}\) equipped with the topology \(\tau_{\mathbb 2} := \{\varnothing,\{1\},\{0,1\}\}\text{.}\) Note that this is not the discrete or concrete topology on \(S\text{.}\) After developing the language of continuity, we will discover that for an arbitrary topological space \(X\text{,}\) the continuous functions \(X\to \mathbb 2\) are in natural bijection with open subsets of \(X\text{.}\)

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Checkpoint 2.1.19. Topologies on a finite set.

Determine all topologies on the set \(\mathbb 2 = \{0,1\}\text{.}\) How many are there? Now do the same thing with \(\mathbb 3 = \{0,1,2\}\text{.}\) (You should draw a picture and it should be large and glorious. How can you organize your picture so that it demonstrates important relations between topologies?)

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It is an open problem to determine the number of topologies on a finite set (as a function of cardinality). You can find known data and references on this problem on the OEIS

oeis.org/A000798

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Example 2.1.20. Cofinite topology.

Given a set \(X\text{,}\) we can define its cofinite topology with open sets

\begin{gather*} \end{gather*}

Note that if \(X\) is finite, then its confinite topology is the same as the discrete topology.

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Example 2.1.21. The Euclidean topology on \(\mathbb R^n\).

We already discussed the open ball basis for the standard (or Euclidean) topology on \(\mathbb R^n\text{.}\) The open sets in this topology are the arbitrary unions of open balls.

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Example 2.1.22. The Zariski topology on \(k^n\).

Fix a field \(k\) and natural number \(n\text{.}\) Given a finite set of polynomials in \(n\) variables \(\mathscr F\subseteq k[x_1,\ldots,x_n]\text{,}\) its vanishing locus is

\begin{gather*} \end{gather*}

The Zariski topology on \(k^n\) is

\begin{gather*} \end{gather*}

In other words, \(U\subseteq k^n\) is Zariski open if and only if there exist finitely many polynomials \(p_i\in k[x_1,\ldots,x_n]\text{,}\) \(i\in I\text{,}\) for which \(x\in U\) if and only if \(p_i(x)\ne 0\) for some \(i\in I\text{.}\)

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Checkpoint 2.1.23. Zariski versus cofinite topologies.

Let \(k\) be a field.

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Show that every cofinite open subset of \(k^n\) is Zariski open.

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Show that when \(n=1\text{,}\) the cofinite and Zariski topologies on \(k\) are the same.

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(c)

Show that for \(k\) infinite and \(n\gt 1\text{,}\) there are Zariski open subsets of \(k^n\) that are not cofinite.

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Example 2.1.24. Poset topology.

Let \((X,\le)\) be a poset. The poset topology on \(X\) has open sets the upwards closed subsets of \(X\text{.}\) This means that \(U\subseteq X\) is open if and only if \(x\in U\text{,}\) \(y\in X\text{,}\) \(x\le y\) implies \(y\in U\text{.}\)

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Example 2.1.25. Subspace topology.

Let \(X\) be a topological space and let \(A\subseteq X\) be a subset. The subspace topology on \(A\) has open sets of the form \(A\cap U\) where \(U\subseteq X\) is open. This is a very powerful construction that allows us to induce topologies on all manner of things. For instance, open intervals \((a,b)\subseteq \mathbb R\) receive a topology in this way, as do open balls \(B = B(p,\varepsilon)\subseteq \mathbb R^n\) and the unit sphere

\begin{gather*} \end{gather*}

If we do not specify otherwise, a subset of a topological space will always be endowed with the subspace topology.

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Here’s a fun way to produce interesting (sub)spaces: first, endow the \(n\times n\) real matrices \(\mathbb R^{n\times n}\) with the Euclidean topoology (considering a matrix \((x_{ij})_{1\le i,j\le n}\) as the vector \((x_{11},\ldots,x_{1n},x_{21},\ldots,x_{2n},\ldots, x_{n1},\ldots,x_{nn})\)). Inside of \(\mathbb R^{n\times n}\) we have the matrix groups

\begin{gather*} \end{gather*}

of special orthogonal, orthogonal, and invertible (general linear) matrices. We will consider each as a subspace of \(\mathbb R^{n\times n}\)k, producing a rich family of topological spaces.

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As a special case, consider \(\mathrm{SO}_2(\mathbb R)\text{.}\) These matrices encode rotations about the origin in the plane. Once we have introduced the concepts of continuity and homeomorphism, we will argue that \(\mathrm{SO}_2(\mathbb R)\) is "the same as" (homeomorphic to) \(S^1\text{,}\) i.e.

\begin{gather*} \end{gather*}

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Subsection Wednesday

Subsubsection Continuity

Given sets \(X,Y\text{,}\) we are familiar with functions \(f\colon X\to Y\) as a mechanism for comparing \(X\) and \(Y\text{.}\) If \(X\) and \(Y\) are topological spaces, then only some functions "respect" their topology; we call these continuous functions. Before introducing the formal definition of continuity, let’s motivate it in terms of familiar \(\varepsilon\)-\(\delta\) continuity:

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A function \(f\colon \mathbb R\to \mathbb R\) is continuous at a point \(p\) in its domain :iff

for all \(\varepsilon \gt 0\) there exists \(\delta \gt 0\) such that \(|x-p|\lt\delta\) implies \(|f(x)-f(p)|\lt\varepsilon\text{.}\)
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We are currently interested in functions continuous everywhere, so the above condition must hold for all \(p\in \mathbb R\text{.}\) Of course, \(\delta\) is permitted to depend on both \(p\) and \(\varepsilon\text{.}\)

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Phrased in terms of the basic open sets of \(\mathbb R\text{,}\) that is, open balls, we see that for every \(p\in \mathbb R\) and every open ball \(B\) centered at \(f(p)\text{,}\) there exists an open ball \(C\) centered at \(p\) such that \(fC\subseteq B\text{;}\) equivalently, \(C\subseteq f^{-1}B\text{.}\)

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We can further recast \(\varepsilon\)-\(\delta\) continuity as follows:

for every \(p\) in the domain and every basic open \(B\) of the codomain with \(f(p)\in B\text{,}\) there exists a basic open \(C\) of the domain such that \(p\in C\) and \(C\subseteq f^{-1}B\text{.}\)
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Now consider some \(x\in f^{-1}B\text{.}\) By continuity, there exists a basic open \(C_x\) of the domain for which \(x\in C_x\) and \(C_x\subseteq f^{-1}B\text{.}\) From this, we may conclude that

\begin{gather*} \end{gather*}

(You should check the details.) Thus \(f^{-1}B\) is open. Observe that every open set is a union of basic opens, whence we get another equivalent condition for continuity:

for every basic open \(B\) of the codomain, \(f^{-1}B\) is open in the domain.
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And once again, every open is a union of basic opens, and preimage commutes with unions, we we can simply say that the preimage of every open is open. This is our formal definition of continuity in the context of general topological spaces:

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Definition 2.1.26. Continuity.

A function \(f\colon X\to Y\) between topological spaces is continuous :iff for every open subset \(V\subseteq Y\text{,}\) the preimage \(f^{-1}V\) is open in \(X\text{.}\)

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In order to develop our intuition for continuity, we’ll introduce some additional terminology:

Definition 2.1.27. Closed subsets.

A subset \(C\subseteq X\) of a topological space \(X\) is closed :iff \(C = X\smallsetminus U\) for some open set \(U\subseteq X\text{.}\)

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Checkpoint 2.1.28. Topologies via closed sets.

Prove that the closed subsets of a topological space are closed under arbitrary intersection and finite union. Then verify that a family of subsets \(\kappa \subseteq 2^X\) closed under arbitrary intersection and finite union specifies a topology by declaring \(U\subseteq X\) to be open if and only if it is the complement of some set in \(\kappa\text{.}\)

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Since \(\varnothing\) and \(X\) are always open subsets of a topological space \(X\text{,}\) we see that \(X\) and \(\varnothing\) are always closed subsets of \(X\text{.}\) In this fashion, we see that "open" and "closed" are not antonyms in topology — they are properties that can be simultaneously obtained. When a set is both open and closed, we may exercise questionable taste and call it clopen.

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Definition 2.1.29. Interior and closure.

Let \(X\) be a space and \(A\subseteq X\) be a subset. The interior of \(A\text{,}\) denoted \(A^\circ\text{,}\) is the maximal open subset of \(A\text{;}\) the closure of \(A\text{,}\) denoted \(\bar A\text{,}\) is the minimal closed subset of \(X\) containing \(A\text{.}\)

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Checkpoint 2.1.30. Constructing interiors and closures.

Prove that for \(A\) a subset of a space \(X\text{,}\)

\begin{gather*} \end{gather*}

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Checkpoint 2.1.31. Continuity in terms of interiors and closures.

Show that a function \(f\colon X\to Y\) between topological spaces in continuous if and only if for all subsets \(B\subseteq Y\text{,}\) \(f^{-1}[B^\circ]\subseteq [f^{-1}B]^\circ\text{.}\) Show that continuity is also equivalent to the condition that for all subsets \(A\subseteq X\text{,}\) \(f\bar A\subseteq \overline{fA}\text{.}\)

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With a good notion of continuity in hand, we are now ready to specify what it means for two spaces to be the "same" --- structurally if not literally.

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Definition 2.1.32. Homeomorphism.

Given topological spaces \(X,Y\text{,}\) a continuous function \(f\colon X\to Y\) is a homeomorphism :iff it has a continuous two-sided inverse \(g\colon Y\to X\text{.}\) Two spaces \(X,Y\) are called homeomorphic :iff a homeomorphism \(f\colon X\to Y\) exists; in this case, we write \(X\cong Y\) or \(f\colon X\xrightarrow{\cong}Y\) or \(f\colon X\cong Y\text{.}\)

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Here is an important theorem that will help you solve the ensuing exercise.

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Theorem 2.1.33. Compositions of continuous functions are continuous.

Suppose \(f\colon X\to Y\) and \(g\colon Y\to Z\) are continuous functions between topological spaces. Then the composite function \(g\circ f\colon X\to Z\) is continuous as well.

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Proof.

Suppose \(U\subseteq Z\) is open. Since \(g\) is continuous, \(g^{-1}U\) is an open subset of \(Y\text{.}\) Since \(f\) is continuous, \(f^{-1}[g^{-1}U]\) is an open subset of \(X\text{.}\) One may check that

\begin{gather*} \end{gather*}

This shows that \((g\circ f)^{-1}U\) is an open subset of \(X\text{.}\) Since \(U\) was an arbitrary open subset of \(Z\text{,}\) we deduce that \(g\circ f\) is continuous.

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Checkpoint 2.1.34. Homeomorphism is an equivalence relation.

Momentarily pretend there is such a thing as the "set" \(\mathrm{Top}\) of all topological spaces. Prove that homeomorphism \(\cong\) is an equivalence relation on \(\mathrm{Top}\text{.}\)

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It is now high time for some examples.

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Example 2.1.35. Identities are homeomorphisms.

The identity function \(\operatorname{id_X}\colon (X,\tau)\to (X,\tau)\) is a homeomorphism. This is because \(\operatorname{id_X}\) is continuous and its inverse is \(\operatorname{id}_X\text{.}\)

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Example 2.1.36. Every function from a discrete space is continuous.

Write \((X,\tau_{\text{disc}})\) for a set \(X\) endowed with the discrete topology \(\tau_{\text{disc}} = 2^X\text{.}\) If \((Y,\tau)\) is some other topological space and \(f\colon X\to Y\) is an arbitrary function, then \(f\) is continuous. Indeed, if \(V\subseteq Y\) is open, then \(f^{-1}V\subseteq X\) is open since all subsets of a discrete space are open.

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Example 2.1.37. Continuous bijections need not be homeomorphisms.

For a set \(X\text{,}\) consider the identity function

\begin{gather*} \end{gather*}

This is a continuous bijection, but its inverse

\begin{gather*} \end{gather*}

is not continuous unless \(X=\varnothing\text{.}\) (Check this!)

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For a more geometric example, consider the function

\begin{align*} [0,1) \amp \longrightarrow S^1 \subseteq \mathbb C \\ t \amp \longmapsto \operatorname{exp}(2\pi i t). \end{align*}

This is a continuous bijection, but the inverse rends the circle at \(1\text{.}\) To see why this fails the formal definition of continuity, note that \([0,1/2)\subseteq [0,1)\) is open, but its preimage \(\{z\in S^1\mid \operatorname{Re}(z)\gt 0\}\cup \{1\}\) is not open.

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Example 2.1.38. Open intervals and \(\mathbb R\).

The open interval \((0,1)\subseteq \mathbb R\) is homemorphic to every other open interval \((a,b)\subseteq \mathbb R\) with \(a\lt b\) via

\begin{align*} (0,1) \amp \longrightarrow (a,b)\\ t \amp \longmapsto a+(b-a)t \end{align*}

which is continuous with continuous inverse

\begin{align*} (a,b) \amp \longrightarrow (0,1)\\ t \amp \longmapsto \frac{t-a}{b-a} \end{align*}

All open intervals are also homeomorphic to \(\mathbb R\text{.}\) Indeed,

\begin{align*} (-1,+1) \amp \longleftrightarrow \mathbb R \\ t \amp \longmapsto \frac{t}{1-t^2}\\ \frac{-1+\sqrt{1+4t^2}}{2t} \amp ⟻ t \end{align*}

are inverse continuous functions (where for the second map, we define \(0\mapsfrom 0\)).

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Subsubsection Topological properties

Historically, topological properties preceded topology. Mathematicians noticed that certain properties — such as nonemptiness, connectedness, discreteness, and compactness — did not change under a broad class of deformations. In order codify these properties, they then created the language of topology, continuity, and homeomorphisms.

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Definition 2.1.39. Topological property.

Let \(P\) be a property that may or may not be obtained by each topological space. Call \(P\) a topological property :iff

\begin{gather*} \end{gather*}

In other words, a property is a topological property if and only if it is homeomorphism invariant.

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Example 2.1.40. Cardinality is a topological property.

If \(X\cong Y\text{,}\) then a homeomorphism \(f\colon X\xrightarrow{\cong}Y\) is necessarily a bijection, whence the cardinalities of \(X\) and \(Y\) are equal: \(\#X=\#Y\text{.}\)

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Checkpoint 2.1.41. Cardinality of the topology is a topological property.

Prove that \((X,\tau)\cong (X',\tau')\) implies \(\#\tau=\#\tau'\text{.}\)

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Much of this course will develop additional topological properties, such as (path) connectivity, connectedness, the (isomorphism type of the) fundamental group, and the (isomorphism type of) homology.

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Subsubsection Building continuous functions

Our task now is to assemble a toolkit that will allow us to build new continuous functions from old ones. These notes will prove a few of the statements, you will prove a few in your homework, and the rest will be left for personal verification or faith. Recall that in Theorem 2.1.33 we have already verified that composition preserves continuity, and in Example 2.1.35 we learned that identity functions (from a space to the same space) are continuous.

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Theorem 2.1.42. New continuous maps from old.

Let \(X,Y\) be topological spaces.

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Constant maps 🔗: Fix \(y_0\in Y\text{.}\) Then the constant map \(\operatorname{const}_{y_0}\colon X\to Y,~x\mapsto y_0\) is continuous.
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Open inclusions 🔗: Let \(U\subseteq X\) be an open subset. The inclusion map \(\operatorname{inc}\colon U\hookrightarrow X,~x\mapsto x\) is continuous.
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Into an open 🔗: Let \(V\subseteq Y\) be open and let \(f\colon X\to V\) be a function. If the composite \(\operatorname{inc}\circ f\colon X\to V\to Y\) is continuous, then so is \(f\text{.}\)
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Check on closed sets 🔗: A function \(f\colon X\to Y\) is continuous if and only if the preimage of every closed subset of \(Y\) is closed.
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Locality 🔗: Let \(f\colon X\to Y\) be a function. Suppose that for each \(x\in X\) there exists an open subset \(x\in U_x\subseteq X\) for which \(f\circ \operatorname{inc}\colon U_x\hookrightarrow X\to Y\) is continuous. Then \(f\) is continuous.
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Gluing 🔗: Suppose \(X=A\cup B\text{,}\) where \(A\) and \(B\) are both open subsets (or, alternatively, both closed subsets) of \(X\text{.}\) Give \(A\) and \(B\) the subspace topology and suppose that \(f\colon X\to Y\) is a function whose restrictions to \(A\) and \(B\) are continuous. Then \(f\) is continuous.
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Proof.

We will only verify locality and the open case of gluing and leave the other properties as exercises. Locality first: Suppose that \(f\colon X\to Y\) satisfies the hypotheses of locality and let \(V\subseteq Y\) be an open subset. Then

\begin{gather*} \end{gather*}

is open for all \(x\in X\text{.}\) Furthermore,

\begin{gather*} \end{gather*}

since the \(U_x\) cover \(X\text{.}\) Since arbitrary unions of open sets are open, we conclude that \(f^{-1}V\) is open. Since \(V\subseteq Y\) was an arbitrary open subset, this demonstrates that \(f\) is continuous.

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Now for gluing when \(A,B\subseteq X\) are open: Let \(V\subseteq Y\) be open. Then —(by hypothesis and the definition of the subspace topology — \(A\cap f^{-1}V\) is open in \(A\) and \(B\cap f^{-1}V\) is open in \(B\text{.}\) Thus \(A\cap f^{-1}V = U_A\cap A\) for some open set \(U_A\subseteq X\) and \(B\cap f^{-1}V = U_B\cap B\) for some open set \(U_B\subseteq X\text{;}\) as the finite intersection of open subsets of \(X\text{,}\) we conclude that \(A\cap f^{-1}V\) and \(B\cap f^{-1}V\) are open in \(X\text{.}\) Since \(X=A\cup B\text{,}\) we see that

\begin{gather*} \end{gather*}

is the union of open subsets of \(X\) and hence open. Since \(V\subseteq Y\) was an arbitrary open subset, this verifies that \(f\) is continuous.

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Many of our constructions of topological spaces will come equipped with methods for identifying continuous maps either in or out of the space. For the moment, we will content ourselves with a discussion of how one might verify continuity from general principles instead of via the raw definition.

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Checkpoint 2.1.43. Breaking down continuity.

Suppose you want to show that

\begin{align*} f\colon \mathbb R^2 \amp \longrightarrow \mathbb R^2 \\ (x,y) \amp \longmapsto (xy,y+2) \end{align*}

is continuous. Come up with a roadmap of general lemmata you might try to prove in service of deducing continuity of this map. We will discuss your roadmap in class and determine if it can work.

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Hint.

You probably want addition and multiplication to be continuous. And it would also be nice if continuity after projection to each factor implied continuity of the original map.

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Checkpoint 2.1.44. Build your own continuous function.

Define an interesting function \(S^1\to\mathbb R\) (or perhaps with a different codomain). Come up with a criterion that you hope will test whether your function is continuous in terms of a related function from a closed interval to \(\mathbb R\) (or your alternate codomain).

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Hint.

What can you do with \([0,1]\to S^1\text{,}\) \(t\mapsto \operatorname{exp}(2\pi i t)\text{?}\)

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Once we develop the product and quotient topologies, we will be able to vet your guesses in the above exercises.

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Subsection Friday

We finished up most of the Wednesday material during this class meeting.

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