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Worksheet 3.6 Homework 06
Instructions: Complete all the exercises below and submit your work via
Gradescope by Monday March 30 at 10pm.
1. Recall that
\(S^0 = \{\pm 1\}\) carries the discrete topology. For an arbitrary space
\(X\text{,}\) construct a bijection between continuous surjective maps
\(X\to S^0\) and separations of
\(X\text{.}\) (Prove that your assignment is a bijeciton.)
2. Use a construction involving connected components to prove that
\(\mathbb R^2\) is not homeomorphic to
\(\mathbb R\text{.}\)
3. Let
\(n\) be a positive integer and let
\(f\colon S^n\to\mathbb R\) be a continuous map. Prove that there is a point
\(p\in S^n\) satisfying
\(f(p)=f(-p)\text{.}\)
4. Prove that
\(\#\pi_0\operatorname{Conf}_n\mathbb R = n!\text{.}\)
5. Let
\(X\) be a topological space.
(a)
Prove that the map
\begin{equation*}
\coprod_{U\in [X]} U\longrightarrow X,\qquad x\longmapsto x
\end{equation*}
is a continuous bijection.
(b) Prove that, if
\(X\) is locally connected, then this map is a homeomorphism.
(c) Give an example of an
\(X\) for which this map is not a homeomorphism.
