J. Munkres. Topology (2013) Chapter 4: Countability and Separability Axioms. 30: The Countability Axioms

Definition

A topological space is second countable iff it has a countable basis.

Example 1

\(\mathbb{R}\), \(\mathbb{R}^n\) for \(n \in \mathbb{Z}_+\) and \(\mathbb{R}^\omega\) are all second countable.

Proposition

In a second countable space, any discrete subspace is countable.

Proof

• Let \(X\) be a second countable space.
• Pick a countable basis \(\mathcal{B}\).
• Let \(A \subseteq X\) be discrete.
• For all \(a \in A\), pick \(B_a \in \mathcal{B}\) such that \(B_a \cap A = \{a\}\).
• The function that maps \(a\) to \(B_a\) is an injection \(A \rightarrow \mathcal{B}\).
• \(A\) is countable.
• \(\Box\)

Example 2

\(\mathbb{R}^\omega\) under the uniform topology is not second countable. The set of sequences consisting of all 0s and 1s is an uncountable discrete subspace.

Theorem 30.2

A subspace of a first countable space is first countable. A countable product of first countable spaces is first countable. A subspace of a second countable space is secound countable. A countable product of second countable spaces is second countable.

Proof

• A subspace of a first countable space is first countable.
• Let \(X\) be a first countable space.
• Let \(Y \subseteq X\).
• Let \(y \in Y\).
• Pick a countable basis \(\mathcal{B}\) at \(y\) in \(X\). Prove: \(\{ B \cap Y \mid B \in \mathcal{B} \}\) is a basis at \(y\) in \(Y\).
• Let \(U\) be a neighbourhood of \(y\) in \(Y\).
• Pick a neighbourhood \(V\) of \(y\) in \(X\) such that \(U = V \cap Y\).
• Pick \(B \in \mathcal{B}\) such that \(B \subseteq V\)
• \(B \cap Y \subseteq U\)
• A countable product of first countable spaces is first countable.
• Let \((X_n)\) be a sequence of first countable spaces.
• Let \((x_n) \in \prod_n X_n\)
• For \(n \in \mathbb{Z}_+\), pick a countable basis \(\mathcal{B}_n\) at \(x_n\) in \(X_n\)
• Let \(\mathcal{B}\) be the set of all sets of the form \(\prod_n U_n\) where \(U_n \in \mathcal{B}_n\) for finitely many n, and \(U_n = X_n\) for all other \(n\).
• \(\mathcal{B}\) is a countable basis at \((x_n)\) in \(\prod_n X_n\).
• A subspace of a second countable space is second countable.
• Let \(X\) be a second countable space.
• Let \(Y \subseteq X\).
• Pick a countable basis \(\mathcal{B}\) for \(X\).
• \(\{B \cap Y \mid B \in \mathcal{B}\}\) is a countable basis for \(Y\).
• A countable product of second countable spaces is second countable.
• Let \((X_n)\) be a sequence of second countable spaces.
• For each \(n\), pick a countable basis \(\mathcal{B}_n\) for \(X_n\).
• Let \(\mathcal{B}\) be the set of all sets of the form \(\prod_n U_n\), where \(U_n \in \mathcal{B}_n\) for finitely many \(n\), and \(U_n = X_n\) for all other \(n\).
• \(\mathcal{B}\) is a countable basis for \(\prod_n X_n\).
• \(\Box\)

Definition

A subset \(A\) of a topological space \(X\) is dense iff \(\overline{A} = X\).

Definition

A topological space is Lindelöf iff every open covering has a countable subcovering.

Definition

A topological space is separable iff there exists a countable dense subset.

Theorem 30.3

Suppose that \(X\) is second countable. Then:

1. \(X\) is Lindelöf.
2. \(X\) is separable.

Proof

• Pick a countable basis \(\mathcal{B}\) for \(X\).
• Every open covering of \(X\) has a countable subcovering.
• Let \(\mathcal{U}\) be an open covering of \(X\).
• For each \(B \in \mathcal{B}\) such that \(\exists U \in \mathcal{U}. B \subseteq U\), pick \(U_B \in \mathcal{U}\) such that \(B \subseteq U\). Prove: \(\{U_B \mid B \in \mathcal{B}\}\) covers \(X\).
• Let \(x \in X\).
• Pick \(U \in \mathcal{U}\) such that \(x \in U\).
• Pick \(B \in \mathcal{B}\) such that \(x \in B \subseteq U\).
• \(x \in U_B\)
• There exists a countable subset of \(X\) that is dense in \(X\).
• For \(B \in \mathcal{B}\) nonempty, pick \(x_B \in B\). Prove: \(D = \{x_B \mid B \in \mathcal{B}\}\) is dense.
• Let \(U\) be a nonempty open set in \(X\). Prove: \(U\) intersects \(D\).
• Pick \(x \in U\)
• Pick \(B \in \mathcal{B}\) such that \(x \in B \subseteq U\).
• \(x_B \in U \cap D\)
• \(\Box\)

Example 3

The space \(\mathbb{R}_l\) is first countable, Lindelöf and separable, but not second countable.

• \(\mathbb{R}_l\) is first countable.
• For any \(x \in \mathbb{R}\), the set \(\{[x,x+1/n) \mid n \in \mathbb{Z}_+\}\) is a countable basis at \(x\).
• \(\mathbb{R}_l\) is Lindelöf.
• \( \langle 2 \rangle 1 \) Any covering of \(\mathbb{R}_l\) by basis elements has a countable subcovering.
• Let \(\mathcal{A}\) be a set of basis elements that covers \(\mathbb{R}_l\)
• Let \(C = \bigcup \{ (a,b) \mid [a,b) \in \mathcal{A}\}\)
• \(\mathbb{R} - C\) is countable.
• Pick a countable \(\mathcal{A}' \subseteq \mathcal{A}\) that covers \(\mathbb{R} - C\).
• For every point \(x \in \mathbb{R}-C\), pick \(A_x \in \mathcal{A}\) such that \(x \in A_x\).
• Take \(\mathcal{A}' = \{A_x \mid x \in \mathbb{R} - C\}\).
• \(C\) under the standard topology is second countable.
• \(\{(a,b) \mid [a,b) \in \mathcal{A} \}\) is an open covering of \(C\) in the standard topology.
• Pick a countable subset \(\{ (a_1,b_1), (a_2,b_2), \ldots \}\) that covers \(C\).
• \(\mathcal{A}' \cup \{ [a_1,b_1), [a_2,b_2), \ldots \}\) is a finite subset of \(\mathcal{A}\) that covers \(\mathbb{R}\).
• Let \(\mathcal{U}\) be an open covering of \(\mathbb{R}_l\).
• Let \(\mathcal{V}\) be the set of all basis elements \(B\) such that \(\exists U \in \mathcal{U}. B \subseteq U\).
• \(\mathcal{V}\) covers \(\mathbb{R}_l\)
• Pick a countable subcovering \(\mathcal{V}_0\)
• By \(\langle 2 \rangle 1 \).
• For \(V \in \mathcal{V}_0\), pick \(U_V \in \mathcal{U}\) such that \(V \subseteq U_V\)
• \(\{U_V \mid V \in \mathcal{V}\}\) is a countable subset of \(\mathcal{U}\) that covers \(\mathbb{R}_l\).
• \(\mathbb{R}_l\) is separable.
• Since \(\mathbb{Q}\) is dense.
• \(\mathbb{R}_l\) is not second countable.
• Let \(\mathcal{B}\) be any basis for \(\mathbb{R}_l\).
• For \(x \in \mathbb{R}\), pick \(B_x \in \mathcal{B}\) such that \(x \in B_x \subseteq [x,x+1)\).
• The function that maps \(x\) to \(B_x\) is an injective map \(\mathbb{R} \rightarrow \mathcal{B}\).
• If \(x \neq y\), say \(x < y\), then \(x \in B_x\) and \(x \notin B_y\), so \(B_x \neq B_y\).
• \(\mathcal{B}\) is uncountable.
• \(\Box\)

Definition

The Sorgenfrey plane is \(\mathbb{R}_l^2\).

Example 4

The Sorgenfrey plane is not Lindel&omul;f. This shows that the product of two Lindelöf spaces is not necessarily Lindelöf.

• Let \(L = \{(x,-x) \mid x \in \mathbb{R}\}\).
• \(L\) is closed in \(\mathbb{R}_l^2\).
• \(\mathcal{U} = \{\mathbb{R}_l^2 - L\} \cup \{[a,b) \times [-a,d) \mid a,b,d \in \mathbb{R}\}\) is an open covering of \(\mathbb[R}_l^2\).
• \(L\) intersects each element of \(\mathcal{U}\) in at most one point.
• \(L\) is uncountable.
• No countable subset of \(\mathcal{U}\) covers \(L\).
• \(\Box\)

Example 5

A subspace of a Lindelöf space is not necessarily Lindelöf.

The ordered square is compact, hence Lindelöf; but the subspace \([0,1] \times (0,1)\) is not, since the open covering \(\{\{x\} \times (0,1) \mid x \in [0,1]\}\) has no countable subcovering.

Exercises

Exercise 1

(a)

A \(G_\delta\) set in a topological space \(X\) is a set \(A\) that is a countable intersection of open sets of \(X\). Show that in a first countable \(T_1\) space, every one-point set is a \(G_\delta\) set.

Solution

• Let \(x \in X\). Prove: \(\{x\}\) is \(G_\delta\).
• Pick a countable basis \(\mathcal{B}\) at \(x\). Prove: \(\bigcap \mathcal{B} = \{x\}\).
• Assume for a contradiction \(y \in \bigcap \mathcal{B}\) and \(y \neq x\).
• \(x \in X - \{y\}\) and \(X - \{y\}\) is open.
• Pick \(B \in \mathcal{B}\) such that \(B \subseteq X - \{y\}\)
• \(y \in B\)
• This is a contradiction.
• \(\Box\)

(b)

There is a familiar space in which every one-point set is a \(G_\delta\) set, which nevertheless does not satisfy the first countability axiom. What is it?

Solution

\(\mathbb{R}^\omega\) in the box topology is not first countable (Example 21.1), but every singleton \(\{(x_n)\}\) is equal to \(\bigcap_{m \in \mathbb{Z}_+} \prod_n (x_n - 1/m, x_n + 1/m) \).

Exercise 2

Show that in a second countable tapological space, every basis includes a countable basis.

Solution

• Assume \(X\) is second countable.
• Pick a countable basis \(\mathcal{B}\) for \(X\).
• Let \(\mathcal{C}\) be any basis for \(X\).
• For every pair of sets \(B, B' \in \mathcal{B}\) such that there exists \(C \in \mathcal{C}\) such that \(B \subseteq C \subseteq B'\), choose \(C_{BB'} \in \mathcal{C}\) such that \(B \subseteq C_{BB'} \subseteq B'\). Prove: the set of all \(C_{BB'}\) is a basis for \(X\).
• Let \(U\) be any open set in \(X\) and \(x \in U\).
• Pick \(B' \in \mathcal{B}\) such that \(x \in B'\).
• Pick \(C \in \mathcal{C}\) such that \(x \in C \subseteq B'\).
• Pick \(B \in \mathcal{B}\) such that \(x \in B \subseteq C\).
• \(x \in C_{BB'}\).
• \(\Box\)

Exercise 3

In a second countable space, every uncountable set contains uncountably many of its own limit points.

Solution

• Let \(X\) be a second countable space.
• Pick a countable basis \(\mathcal{B}\) for \(x\).
• Let \(A \subseteq X\) be uncountable.
• For every \(x \in A\) that is not a limit point of \(A\), pick \(B_x \in \mathcal{B}\) such that \(B_x \cap A = \{x\}\).
• The function that maps \(x\) to \(B_x\) is an injection.
• Only countably many points of \(A\) are not limit points of \(A\).
• Uncountably many points of \(A\) are limit points of \(A\).
• \(\Box\)

Exercise 4

Every compact metrizable space is second countable.

Solution

• Let \(X\) be a compact metric space.
• For \(n \in \mathbb{Z}_+\), pick a finite covering \(\mathcal{A}_n\) of \(X\) be \(1/n\)-balls.
• Since the set of all \(1/n\)-balls is an open covering of \(X\).
• Let \(\mathcal{B} = \bigcup_n \mathcal{A}_n\).
• Let \(U\) be an open set in \(X\) and \(x \in U\).
• Pick \(\epsilon > 0\) such that \(B(x,\epsilon) \subseteq U\).
• Pick \(n\) such that \(2/n \leq \epsilon\).
• Pick \(B \in \mathcal{A}_n\) such that \(x \in B\).
• \(B \subseteq B(x,\epsilon) \subseteq U\)
• \(\Box\)

Exercise 5

(a)

Every separable metrizable space is second countable.

Solution

• Let \(X\) be a separable metric space.
• Pick a countable dense set \(D\).
• Let \(\mathcal{B} = \{B(d,1/n) \mid d \in D, n \in \mathbb{Z}_+\}\).
• \(\mathcal{B}\) is a basis for \(X\).
• Let \(U\) be an open set in \(X\) and \(x \in U\).
• Pick \(\epsilon > 0\) such that \(B(x,\epsilon) \subseteq U\).
• Pick \(n \in \mathbb{Z}_+\) such that \(2/n \leq \epsilon\).
• Pick \(d \in B(x,1/n) \cap D\)
• \(x \in B(d,1/n) \subseteq B(x,2/n) \subseteq B(x,\epsilon) \subseteq U\)
• \(\Box\)

(b)

Every Lindelöf metrizable space is second countable.

Solution

• Let \(X\) be a Lindel&omul;f metrizable space.
• For \(n \in \mathbb{Z}_+\), let \(\mathcal{A}_n\) be a countable set of \(1/n\)-balls that covers \(X\).
• Let \(\mathcal{B} = \bigcup_n \mathcal{A}_n\).
• Let \(U\) be an open set in \(X\) and \(x \in U\).
• Pick \(\epsilon > 0\) such that \(B(x,\epsilon) \subseteq U\).
• Pick \(n\) such that \(2/n \leq \epsilon\).
• Pick \(B \in \mathcal{A}_n\) such that \(x \in B\).
• \(B \subseteq B(x,\epsilon) \subseteq U\)
• \(\Box\)

Exercise 6

Show that \(\mathbb{R}_l\) and \(I_o^2\) are not metrizable.

Solution

We know \(\mathbb{R}_l\) is not metrizable because it is separable and Lindelöf but not second countable (Example 3).

We know \(I_o^2\) is compact. The subspace \(A = [0,1] \times (0,1)\) is not Lindelöf (Example 5), therefore not second countable (Theorem 30.3), therefore \(I_o^2\) is not second countable (Theorem 30.2), therefore not metrizable (Exercise 4).

Exercise 7

Which of our four countability axioms does \(\Omega\) satisfy? What about \(\Omega + 1\)?

Solution

\(\Omega\) is first countable. For any \(\alpha \neq 0 \in \Omega\), we have \(\{(\beta, \alpha + 1) \mid \beta < \alpha\}\) is a basis at \(\alpha\). And \(\{\{0\}\}\) is a basis at \(0\).

\(\Omega\) is not separable. Let \(D \subseteq \Omega\) be countable. Let \(\alpha = \sup D\). Then \((\alpha, +\infty)\) is a nonempty open set that does not intersect \(D\).

Therefore, \(\Omega\) is not second countable.

\(\Omega\) is not Lindelöf. The open covering \(\{(-\infty, \alpha) \mid \alpha \in \Omega\}\) has no countable subcovering.

\(\Omega + 1\) is not first countable. Assume for a contradiction \(\mathcal{B}\) is a countable basis at \(\Omega\). For \(B \in \mathcal{B}\), pick \(\alpha_B < \Omega\) such that \((\alpha_B, \Omega] \subseteq B\). Let \(\alpha = \sup_{B \in \mathcal{B}} \alpha_B\). Then \(\alpha < \Omega\) since it is the supremum of countably many countable ordinals. There is no \(B \in \mathcal{B}\) such that \(B \subseteq (\alpha + 1, \Omega]\). This is a contradiction.

\(\Omega + 1\) is not separable. Let \(D \subseteq \Omega + 1\) be countable. Let \(\alpha = \sup (D - \{\Omega\})\). Then \((\alpha, \Omega)\) is a nonempty open set that does not intersect \(D\).

Therefore, \(\Omega + 1\) is not second countable.

\(\Omega + 1\) is compact (Theorem 27.1) hence Lindelöf.

Exercise 8

Which of our four countability axioms does \(\mathbb{R}^\omega\) in the uniform topology satisfy?

Solution

\(\mathbb{R}^\omega\) is first countable because it is metrizable. It is not second countable (Example 2). Therefore it is not separable or Lindel&omul;f (Exercise 5).

Exercise 9

Let \(A\) be a closed subspace of \(X\). Show that if \(X\) is Lindelöf, then \(A\) is Lindelöf. Show by example that if \(X\) is separable, then \(A\) need not be separable.

Solution

• Assume \(X\) is Lindelöf
• Let \(\mathcal{U}\) be any open covering of \(A\).
• \{\{V \text{ open in } X \mid V \cap A \in \mathcal{U}\} \cup \{X - A\}\) is an open covering of \(X\).
• Pick a countable subcovering \(\{V_1, V_2, \ldots\}\) or \(\{V_1, V_2, \ldots \} \cup \{ X - A\}\).
• \(\{V_1 \cap A, V_2 \cap A, \ldots \}\) is a countable subset of \(\mathcal{U}\) that covers \(A\).
• \(\Box\)

The Sorgenfrey plane is separable; \(\mathbb{Q}^2\) is a countable dense subset. But the subspace \(L\) from Example 4 is uncountable and discrete, and so is not separable.

Exercise 10

Show that a countable product of separable spaces is separable.

Solution

• Let \((X_n)\) be a sequence of separable spaces.
• For \(n \in \mathbb{Z}_+\), choose a countable dense set \(D_n\) in \(X_n\). Prove: \(\prod_n D_n\) is dense in \(\prod_n X_n\).
• Let \(U \subseteq X_n\) be a nonempty open set.
• Pick \(\prod_n U_n \subseteq U\) where each \(U_n\) is open in \(X_n\).
• For \(n \in \mathbb{Z}_+\), choose \(d_n \in D_n \cap U_n\).
• \((d_n) \in \prod_n D_n \cap U\)

Note that the same proof shows that a countable product of separable spaces under the box topology is separable.

Exercise 11

Let \(f : X \rightarrow Y\) be continuous. Show that if \(X\) is Lindelöf or separable, then \(f(X)\) satisfies the same condition.

Solution

• If \(X\) is Lindelöf then \(f(X)\) is Lindelöf.
• Assume \(X\) is Lindelöf
• Let \(\mathcal{V}\) be an open covering of \(f(X)\).
• \(\{f^{-1}(V) \mid V \text{ open in } Y, V \cap f(X) \in \mathcal{V}\}\) is an open covering of \(X\).
• Pick a countable subset \(\{f^{-1}(V_1), f^{-1}(V_2), \ldots\}\).
• \(\{V_1 \cap f(X), V_2 \cap f(X), \ldots\}\) is a countable subset of \(\mathcal{V}\) that covers \(f(X)\).
• If \(X\) is separable then \(f(X)\) is separable.
• Assume \(X\) is separable.
• Pick a countable dense subset \(D\) of \(X\). Prove: \(f(D)\) is dense in \(f(X)\).
• Let \(V\) be a nonempty open subset of \(f(X)\).
• Pick \(V'\) open in \(Y\) such that \(V = V' \cap f(X)\).
• Pick \(d \in D \cap f^{-1}(V')\)
• \(f(d) \in f(D) \cap V\)
• \(\Box\)

Exercise 12

Let \(f : X \rightarrow Y\) be a continuous open map. Show that if \(X\) is first countable or second countable, then \(f(X)\) satisfies the same condition.

Solution

• If \(X\) is first countable then \(f(X)\) is first countable.
• Assume \(X\) is first countable.
• Pick \(x \in X\). Prove: there exists a countable basis for \(f(X)\) at \(f(x)\).
• Pick a countable basis \(\mathcal{B}\) for \(X\) at \(x\). Prove: \( \{ f(B) \mid B \in \mathcal{B}\}\) is a basis for \(f(X)\) at \(f(x)\).
• Let \(V\) be a neighbourhood of \(f(x)\) in \(f(X)\).
• Pick an open set \(V'\) in \(Y\) such that \(V = V' \cap f(X)\).
• \(f^{-1}(V) = f^{-1}(V')\) is a neighbourhood of \(x\) in \(X\).
• Pick \(B \in \mathcal{B}\) such that \(B \subseteq f^{-1}(V)\).
• \(f(B) \subseteq V\)
• If \(X\) is second countable then \(f(X)\) is second countable.
• Assume \(X\) is second countable.
• Pick a countable basis \(\mathcal{B}\) for \(X\). Prove: \(\( \{ f(B) \mid B \in \mathcal{B}\}\) is a basis for \(f(X)\).
• Let \(x \in X\) and \(V\) be a neighbourhood of \(f(x)\) in \(f(X)\).
• Pick an open set \(V'\) in \(Y\) such that \(V = V' \cap f(X)\).
• \(f^{-1}(V) = f^{-1}(V')\) is a neighbourhood of \(x\) in \(X\).
• Pick \(B \in \mathcal{B}\) such that \(x \in B \subseteq f^{-1}(V)\).
• \(f(x) \subseteq f(B) \subseteq V\)
• \(\Box\)

Exercise 13

Show that if \(X\) is separable, every set of pairwise disjoint open sets in \(X\) is countable.

Solution

• Assume \(X\) is separable.
• Pick a countable dense subset \(D\) of \(X\).
• Let \(\mathcal{U}\) be a set of pairwise disjoint open sets in \(X\).
• For \(U \in \mathcal{U}\) nonempty, pick \(d_U \in D \cap U\).
• The function that maps \(U\) to \(d_U\) is an injective function \(\mathcal{U} - \{\emptyset\} \rightarrow D\).
• \(\mathcal{U}\) is countable.
• \(\Box\)

Exercise 14

Show that if \(X\) is Lindelöf and \(Y\) is compact, then \(X \times Y\) is Lindelöf.

Solution

• Assume \(X\) is Lindelöf and \(Y\) is compact.
• Let \(\mathcal{U}\) be an open covering of \(X \times Y\).
• \(\{W \text{ open in } X \mid W \times Y \text{ can be covered by finitely many elements of } \mathcal{U}\}\) is an open covering of \(X\).
• Let \(x \in X\)
• \(\{x\} \times Y\) is compact.
• It is homeomorphic to \(Y\).
• Pick finitely many \(U_1, \ldots, U_n \in \mathcal{U}\) that cover \(\{x\} \times Y\).
• There exists \(W\) open in \(X\) such that \(x \in W\) and \(W \times Y \subseteq U_1 \cup \cdots \cup U_n\).
• Tube Lemma
• Pick a countable subcover \(\{W_1, W_2, \ldots,\}\).
• For \(n \in \mathbb{Z}_+\), pick finitely many elements \(U_{n1}, \ldots, U_{nr_n} \in \mathcal{U}\) such that \(W_n \times Y \subseteq U_{n1} \cup \cdots \cup U_{nr_n}\).
• \(\{U_{ni} \mid n \in \mathbb{Z}_+, 1 \leq i \leq r_n\}\) is a countable subset of \(\mathcal{U}\) that covers \(X \times Y\).
• \(\Box\)

Exercise 15

Give \(\mathbb{R}^I\) the uniform metric, where \(I = [0,1]\). Let \(\mathcal{C}(I,\mathbb{R})\) be the subspace consisting of continuous functions. Show that \(\mathcal{C}(I,\mathbb{R})\) is separable, and therefore second countable.

Proof

• Let \(D\) be the set of all continuous functions whose graphs consist of finitely many line segments with rational end points.
• \(D\) is countable.
• Let \(f \in \mathcal{C}(I,\mathbb{R})\) and \(\epsilon > 0\). Prove: \(B(f,\epsilon)\) intersects \(D\).
• \(f\) is uniformly continuous.
• Pick \(\delta > 0\) such that, for all \(x,y \in I\), if \(d(x,y) < \delta\) then \(d(f(x),f(y)) < \epsilon / 4\).
• Pick a finite sequence of rationals \((x_1, \ldots, x_n)\) in \(I\) such that \(x_1 = 0\), \(x_n = 1\), and \(d(x_i,x_{i+1}) < \delta\) for all \(i\).
• For \(i = 1, \ldots, n\), pick a rational \(q_i\) such that \(|f(x_i) - q_i| < \epsilon / 8\).
• For all \(i\) we have \(|q_i - q_{i+1}| < 7 \epsilon / 12\)
\[ \begin{align} |q_i - q_{i+1}| & \leq |q_i - f(x_i)| + |f(x_i) - f(x_{i+1})| + |f(x_{i+1}) - q_{i+1}| \\ & < \epsilon / 8 + \epsilon / 4 + \epsilon / 8 \\ & = \epsilon / 2 \end{align} \]
• Let \(g \in D\) be the function whose graph is the sequence of line segments connecting \((x_1,q_1), (x_2,q_2), \ldots, (x_n,q_n)\). Prove: \(d(f,g) < \epsilon\)
• Let \(t \in I\). Prove: \(|f(t) - g(t)| < 7 \epsilon / 8\)
• Pick \(i\) such that \(q_i \leq t \leq q_{i+1}\)
• \(|f(t) - g(t)| < 7 \epsilon / 8\)
\[ \begin{align} |f(t) - g(t)| & \leq |f(t) - f(x_i)| + |f(x_i) - q_i| + |q_i - g(t)| \\ & \leq |f(t) - f(x_i)| + |f(x_i) - q_i| + |q_i - q_{i+1}| \\ & < \epsilon / 4 + \epsilon / 8 + \epsilon / 2 \\ & = 7 \eplino / 8 \end{align} \]
• \(\Box\)

Exercise 16

(a)

Show that the product space \(\mathbb{R}^I\) where \(I = [0,1]\) is separable.

Solution

• Let \(D\) be the set of all functions \(f\) such that there exists a finite sequence of rationals \(x_1, \ldots, x_n\) such that \(x_1 = 0\), \(x_n = 1\), and \(f\) is constant on \([x_i,x_{i+1})\) for all \(1 \leq i < n\). Prove: \(D\) is dense in \(\mathbb{R}^I\).
• Let \(U\) be a nonempty open set. Prove: \(U\) intersects \(D\).
• Let \(\prod_{x \in I} U_x \subseteq U\) where each \(U_x\) is open in \(\mathbb{R}\), and \(U_x = \mathbb{R}\) except for \(x = x_1, \ldots, x_n\), where \(x_1 < \cdots < x_n\)
• Pick a sequence of rationals \(q_1, \ldots, q_{n+1}\) such that \(q_1 = 0\), \(q_{n+1} = 1\), and \(x_i < q_{i+1} < x_{i+1}\) for all \(i\).
• For \(i = 1, \ldots, n\), pick \(y_i \in U_{x_i}\).
• Define \(f : I \rightarrow \mathbb{R}\) by \(f(t) = y_i\) if \(q_i \leq t < q_{i+1}\), and \(f(1) = y_n\) if \(x_n = 1\), else \(f(1) = 0\).
• \(f \in D \cap \prod_x U_x\)

(b)

Show that if \(|J| > 2^{\aleph_0}\) then the product space \(\mathbb{R}^J\) is not separable.

Solution

• Let \(J\) be a set.
• For any dense set \(D\) in \(\mathbb{R}^J\), we have \(|J| \leq 2^{|D|}\).
• Let \(D\) be a dense set in \(\mathbb{R}^J\).
• Define \(f : J \rightarrow \mathcal{P} D\) by \(f(\alpha) = D \cap \pi_\alpha^{-1}((0,1)) = \{ g \in D \mid g(\alpha) \in (0,1)\} \).
• \(f\) is injective.
• Let \(\alpha, \beta \in J\).
• Assume \(\alpha \neq \beta\). Prove: \(f(\alpha) \neq f(\beta)\).
• Pick \(g \in D \cap \pi_\alpha^{-1}((0,1)) \cap \pi_\beta^{-1}((1,2))\)
• \(g \in f(\alpha)\) and \(g \notin f(\beta)\).
• If \(|J| > 2^{\aleph_0}\) then there is no countable dense set in \(\mathbb{R}^J\).
• \(\Box\)

Exercise 17

Give \(\mathbb{R}^\omega\) the box topology. Let \(\mathbb{Q}^\infty\) denote the subspace consisting of sequences of rationals that end in an infinite string of 0s. Which of our four countability axioms does this space satisfy?

Solution

• \(\mathbb{Q}^\infty\) is not first countable.
• Assume for a contradiction \(\mathcal{B} = \{B_1, B_2, \ldots\}\) is a countable basis at \(\vec{0}\).
• For \(n \in \mathbb{Z}_+\), pick positive real numbers \(a_{n1}, a_{n2}, \ldots\) such that \((-a_{n1},a_{n1}) \times (-a_{n2},a_{n2}) \times \cdots \subseteq B_n\).
• Let \(U = (-a_{11}/2,a_{11}/2) \times (-a_{22}/2,a_{22}/2) \times \cdots\).
• There is no \(n\) such that \(B_n \subseteq U\).
• This is a contradiction.
• Therefore \(\mathbb{Q}^\intfy\) is not second countable.
• \(\mathbb{Q}^\infty\) is separable and Lindelöf
• Because it is countable.
• \(\Box\)

Exercise 18

Let \(G\) be a first countable topological group. Show that if \(G\) is separable or Lindelöf then \(G\) is second countable.

Solution

• Pick a countable basis \(\{B_1, B_2, \ldots\}\) at \(e\).
• If \(G\) is separable then \(G\) is second countable.
• Assume \(G\) is separable.
• Pick a countable dense subset \(D\).
• \(\{dB_n \mid d \in D, n \in \mathbb{Z}_+\}\) forms a basis for \(G\).
• Let \(g \in G\) and \(U\) be a neighbourhood of \(g\).
• Pick \(d \in D \cap U\).
• Pick \(n\) such that \(B_n \subseteq d^{-1}U\).
• \(dB_n \subseteq U\)
• If \(G\) is Lindelöf then \(G\) is second countable.
• Assume \(G\) is Lindelöf.
• For \(n \in \mathbb{Z}_+\), pick a countable set \(C_n\) such that \(\{cB_n \mid c \in C_n\}\) covers \(G\).
• Since \(\{xB_n \mid x \in G\}\) is an open cover of \(G\).
• \(\{cB_n \mid n \in \mathbb{Z}_+, c \in C_n\}\) is a basis for \(G\).
• Let \(g \in G\) and \(U\) be a neighbourhood of \(g\).
• \( \langle 3 \rangle 1 \) Pick a symmetric neighbourhood \(V\) of \(e\) such that \(VV \subseteq g^{-1}U\).
• Pick \(n\) such that \(B_n \subseteq V\).
• Pick \(c \in C_n\) such that \(g \in cB_n\).
• \(cB_n \subseteq U\)
• \( \langle 4 \rangle 1 \) Let \(b \in B_n\)
• \( \langle 4 \rangle 2 \) Pick \(b' \in B_n\) such that \(g = cb'\)
• \(g^{-1}cb \in g^{-1}U\)
\[ \begin{align} g^{-1}cb & = (b')^{-1}c^{-1}cb & (\langle 4 \rangle 2) \\ & = (b')^{-1} b \\ & \in VV & (\langle 3 \rangle 1, \langle 4 \rangle 1, \langle 4 \rangle 2) \\ & \subseteq g^{-1}U & (\langle 3 \rangle 1) \end{align} \]
• \(cb \in U\)
• \(\Box\)