Topology MATH 342 Spring 2026

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Let \(\mathscr B\) denote the collection of open balls in \(\mathbb R^n\) (with respect to the standard distance function). Prove that \(\mathscr B\) is a basis.

Let \(X\) be a set and suppose \(\mathscr B\subseteq 2^X\) is a basis. Let \(\tau\subseteq 2^X\) be the collection of arbitrary unions of elements of \(\mathscr B\text{.}\) Prove that \(\tau\) is a topology on \(X\text{.}\) Further prove that if \(\tau'\supseteq \mathscr B\) is another topology containing \(\mathscr B\text{,}\) then \(\tau\subseteq \tau'\text{.}\) Thus \(\tau\) is the smallest (aka coarsest) topology on \(X\) containing \(\mathscr B\text{,}\) and thus we may call \(\tau\) the topology generated by \(\mathscr B\).

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 \(\tau\text{.}\)

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?)