Problem Sets

If you haven't worked with open sets before or need a refresher, try proving these results. These are all standard results, but please try to prove them without looking them up. Do not turn these in; the first graded problem set will be assigned next week.

When \(X\) is a metric space, we equip it with the metric topology unless otherwise specified.

1. Let \((X,d)\) be a metric space, let \(x\in X\), let \(r>0\). Show that \[B(x,r)=\{y\in X: d(x,y) < r\}\] is an open set.
2. Show that the metric topology on \(X\) is in fact a topology.
3. Give an example of a metric space \(X\) and a sequence of open sets \(U_1,\dots, U_n\) such that \(\bigcap_i U_i\) is not an open set.
4. Let \((X,d)\) be a metric space. Show that every open subset of \(X\) is a union of open balls. (Note that this need not be a countable union of open balls.)
5. Let \((X,\mathscr{S})\), \((Y,\mathscr{T})\) be topological spaces and let \(f\from X\to Y\) be a continuous map. Show that if \(x_n\to a\), then \(f(x_n) \to f(a)\).

Please keep in mind the course's generative AI policy. If you use an LLM in your problem set, you must acknowledge it at the start of your solution to each problem, for instance "I used [LLM] to [edit/generate] [part/most/all] of this solution". At grader discretion, LLM-generated solutions may receive a flat 95% grade (regardless of correctness) and no further feedback.

1. Let \(B_\Q=\{(a,b)\subset \R\mid a,b\in \Q\}\) be the set of open intervals with rational endpoints. Show that \(B_\Q\) is a basis for a topology on \(\R\) and that it generates the standard topology on \(\R\).

2. (based on Munkres, p. 92, #9) Show that the lexicographic order topology on \(\R\times \R\) is equal to the product topology \(\R_d\times \R\), where \(\R_d\) is the discrete topology on \(\R\).

   Conclude that this topology is metrizable (i.e., it is the metric topology for some metric on \(\R\times \R\).

3. We say that a topological space \((X,\mathcal{T})\) is Hausdorff if for every \(x,y\in X\) with \(x\ne y\), there are open sets \(U\) and \(V\) such that \(x\in U\), \(y\in V\), and \(U\cap V = \emptyset\). Show that if \(X\) is finite and Hausdorff, then \(\mathcal{T}\) is the discrete topology.
4. Let \(X = [0,1]^2\), equipped with the lexicographic order topology. Show that every sequence in \(X\) has a convergent subsequence.

Conventions: Unless otherwise specified:

• \(\R^n\) is equipped with the standard topology (equivalently, the metric topology)
• If \(X\) is a topological space and \(Y\subset X\), then \(Y\) is equipped with the subspace topology

1. Show that \(X\) is Hausdorff if and only if the diagonal \[D=\{(x,x)\in X\times X\mid x\in X\}\] of \(X\times X\) is closed.

For 2-4: in class, we stated that if \(X = \bigcup_{i=1}^n K_i\), where the \(K_i\)'s are closed sets, and \(f: X\to Y\) is a function such that \(f|_{K_i}\) is continuous for every \(i\), then \(f\) is continuous.

2. Prove this statement.

3. In general, this fails. For instance, if \(X\) is Hausdorff, \(f:X\to Y\) is not discontinuous, and \(K_x = \{x\}\) for all \(x\in X\), then \(K_x\) is closed for all \(x\), \(X=\bigcup_{x\in X} K_x\), and \(f|_{K_x}\) is continuous for all \(x\), but \(f\) is discontinuous.

   Find countably many closed subsets \(K_1, K_2, \dots \subset \R\) and a discontinuous function \(f: \R \to \R\) such that \(f|_{K_i}\) is continuous for all \(i\).

4. Suppose that \(X\) is the union of a locally finite collection of closed sets. That is, \(X = \bigcup_{\alpha \in A} K_\alpha\), where \(K_\alpha\) is closed for all \(\alpha\) and where for all \(x\in X\), there is a neighborhood \(U\) of \(x\) such that \(U\) intersects only finitely many of the \(K_\alpha\)'s.

   Show that if \(f: X\to Y\) is a function such that \(f|_{K_\alpha}\) is continuous for all \(\alpha\), then \(f\) is continuous.

5. (Munkres, p. 152, #2) Let \(\{A_n\}\) be a sequence of connected subspaces of \(X\) such that \(A_n\cap A_{n+1}\ne \emptyset\) for all \(n\). Show that \(\bigcup_n A_n\) is connected.
6. Suppose that \(U\subset \R\) is an open set. Show that every connected component of \(U\) is an open interval and show that \(U\) is the disjoint union of at most countably many open intervals.