Unit 4: Countability Axioms: First, Second Countable & Separable Spaces

1. Local Countability: The First Countability Axiom

The first countability axiom ensures that local properties around any point can be controlled by a countable sequence of neighborhoods, making sequential arguments sufficient for topology.

Definition 4.1 (Local Basis / Neighborhood Base): Let $(X, \mathcal{T})$ be a topological space and $x \in X$. A collection of open neighborhoods $\mathcal{B}(x)$ of $x$ is called a local basis (or neighborhood base) at $x$ if for every open neighborhood $U$ of $x$, there exists $B \in \mathcal{B}(x)$ such that:

Definition 4.2 (First-Countable Space - $C_1$): A topological space $(X, \mathcal{T})$ is called first-countable (or satisfies the first axiom of countability) if every point $x \in X$ has a countable local basis.

2. Metric Spaces are First-Countable

Theorem 4.1 (First-Countability of Metric Spaces): Every metric space $(X, d)$ is first-countable.

Proof: For any point $x \in X$, consider the countable family of open balls with rational radii:

For any open neighborhood $U$ of $x$, by Definition 1.3 there exists $r > 0$ such that $B(x, r) \subseteq U$. By the Archimedean property, choose $n \in \mathbb{N}$ such that $1/n < r$. Then:

Since $\mathcal{B}(x)$ is indexed by $\mathbb{N}$, it is countable. Thus every point has a countable local basis, so $(X, d)$ is first-countable. $\blacksquare$

3. Sequences in First-Countable Spaces

In general topological spaces, sequences are insufficient to detect closure or continuity (one requires nets or filters). However, in first-countable spaces, sequences suffice!

Theorem 4.2 (Sequential Characterization of Closure): Let $(X, \mathcal{T})$ be a first-countable space and $A \subseteq X$. A point $x \in \operatorname{cl}(A)$ if and only if there exists a sequence $(a_n)_{n=1}^\infty \subseteq A$ such that $a_n \to x$.

1. Global Countability: The Second Countability Axiom

While first-countability is a local condition, second-countability is a powerful global finiteness condition.

Definition 4.3 (Second-Countable Space - $C_2$): A topological space $(X, \mathcal{T})$ is called second-countable (or satisfies the second axiom of countability) if its topology $\mathcal{T}$ admits a countable basis. That is, there exists a countable collection $\mathcal{B} = \{B_n\}_{n=1}^\infty \subseteq \mathcal{T}$ such that every open set $U \in \mathcal{T}$ is a union of members of $\mathcal{B}$.

Example 4.1 (Euclidean Space $\mathbb{R}^n$ is Second-Countable): In $\mathbb{R}^n$, consider the collection of open balls with rational centers and rational radii:

Since $\mathbb{Q}^n$ and $\mathbb{Q}$ are countable, the Cartesian product $\mathbb{Q}^n \times \mathbb{Q}_{> 0}$ is countable. By denseness of $\mathbb{Q}$ in $\mathbb{R}$, every open ball in $\mathbb{R}^n$ is a union of balls in $\mathcal{B}$. Hence $\mathcal{B}$ is a countable basis, proving $\mathbb{R}^n$ is second-countable.

2. Lindelöf's Covering Theorem

Definition 4.4 (Lindelöf Space): A topological space $X$ is called a Lindelöf space if every open covering of $X$ contains a countable subcovering.

Theorem 4.3 (Lindelöf's Theorem): Every second-countable space is Lindelöf.

Proof: Let $X$ be second-countable with countable basis $\mathcal{B} = \{B_n\}_{n=1}^\infty$. Let $\mathcal{U} = \{U_\alpha\}_{\alpha \in I}$ be any arbitrary open cover of $X$: $\bigcup_{\alpha \in I} U_\alpha = X$.

For each point $x \in X$, there exists some $\alpha(x) \in I$ such that $x \in U_{\alpha(x)}$. Since $\mathcal{B}$ is a basis, there exists a basis element $B_{n(x)} \in \mathcal{B}$ such that:

Consider the subcollection of basis elements:

Since $\mathcal{B}^ \subseteq \mathcal{B}$ and $\mathcal{B}$ is countable, $\mathcal{B}^$ is countable:

For each $B_{n_k} \in \mathcal{B}^*$, choose one set $U_{\alpha_k} \in \mathcal{U}$ containing $B_{n_k}$.

We claim that $\{U_{\alpha_k}\}_{k=1}^\infty$ covers $X$: For any $x \in X$, $x \in B_{n(x)} \subseteq U_{\alpha(x)}$. Since $B_{n(x)} \in \mathcal{B}^*$, $B_{n(x)} = B_{n_k}$ for some $k$. Then $x \in B_{n_k} \subseteq U_{\alpha_k}$. Thus $X = \bigcup_{k=1}^\infty U_{\alpha_k}$. We have extracted a countable subcover from the arbitrary open cover $\mathcal{U}$! Therefore, $X$ is a Lindelöf space. $\blacksquare$

1. Separable Spaces

Definition 4.5 (Separable Space): A topological space $(X, \mathcal{T})$ is called separable if it contains a countable dense subset. That is, there exists a countable set $D \subseteq X$ such that:

Equivalently, $D$ intersects every non-empty open set in $X$.

Example 4.2 ($\mathbb{R}^n$ is Separable): $\mathbb{Q}^n$ is countable and dense in $\mathbb{R}^n$, so $\mathbb{R}^n$ is separable.

Example 4.3 (Non-separable Sequence Space $\ell^\infty$): Consider the space $\ell^\infty$ of bounded real sequences with norm $\|x\|_\infty = \sup_n |x_n|$. For each subset $S \subseteq \mathbb{N}$, define the characteristic sequence $\mathbf{1}_S = (s_n)_{n=1}^\infty$ where $s_n = 1$ if $n \in S$ and $0$ otherwise. For any two distinct subsets $S \ne T \subseteq \mathbb{N}$:

Thus, the open balls $B(\mathbf{1}_S, 1/2)$ for all $S \in \mathcal{P}(\mathbb{N})$ are mutually disjoint! The power set $\mathcal{P}(\mathbb{N})$ has cardinality $2^{\aleph_0} = \mathfrak{c}$ (uncountable). If $D$ were any dense subset of $\ell^\infty$, each of these uncountably many disjoint open balls would have to contain at least one point of $D$. This forces $D$ to be uncountable! Therefore, $\ell^\infty$ cannot be separable.

1. The Countability Implication Hierarchy

For general topological spaces, the relationships among countability axioms are strictly directional:

Theorem 4.4 (Second-Countable Implies Separable): If $(X, \mathcal{T})$ is second-countable, then $X$ is separable.

Proof: Let $\mathcal{B} = \{B_n\}_{n=1}^\infty$ be a countable basis for $X$. For each non-empty $B_n \in \mathcal{B}$, choose one point $d_n \in B_n$. Let $D = \{d_n : B_n \ne \emptyset\}$. $D$ is a countable subset of $X$. For any non-empty open set $U \subseteq X$, since $\mathcal{B}$ is a basis, there exists non-empty $B_k \in \mathcal{B}$ with $B_k \subseteq U$. Then $d_k \in B_k \subseteq U$, so $d_k \in D \cap U$. Thus $D$ intersects every non-empty open set, proving $\operatorname{cl}(D) = X$. Hence $X$ is separable. $\blacksquare$

2. Complete Equivalence in Metric Spaces

In metric spaces, the distinction between global countability properties collapses!

Theorem 4.5 (Metric Countability Equivalence): For any metric space $(X, d)$, the following three properties are logically equivalent:

1. $(X, d)$ is Second-Countable ($C_2$).
2. $(X, d)$ is Separable ($S$).
3. $(X, d)$ is Lindelöf ($L$).

Proof ($S \implies C_2$): Let $D = \{d_n\}_{n=1}^\infty$ be a countable dense subset of $(X, d)$. Consider the countable family of open balls:

We claim $\mathcal{B}$ is a basis for $(X, d)$. Let $U$ be open and $x \in U$. There exists $\varepsilon > 0$ such that $B(x, \varepsilon) \subseteq U$. Choose $m \in \mathbb{N}$ such that $1/m < \varepsilon / 2$. Since $D$ is dense, there exists $d_n \in D$ such that $d(x, d_n) < 1/m$. Then:

1. $x \in B(d_n, 1/m)$.
2. For any $y \in B(d_n, 1/m)$: $d(y, x) \le d(y, d_n) + d(d_n, x) < 1/m + 1/m = 2/m < \varepsilon$.

Thus $B(d_n, 1/m) \subseteq B(x, \varepsilon) \subseteq U$. Hence $\mathcal{B}$ is a countable basis, proving $X$ is second-countable. $\blacksquare$

1. The Sorgenfrey Line (Lower Limit Topology)

Robert Sorgenfrey introduced in 1947 one of the most famous counterexamples in general topology.

Definition 4.6 (Sorgenfrey Line $\mathbb{R}_\ell$): The Sorgenfrey line $\mathbb{R}_\ell$ is the real line $\mathbb{R}$ equipped with the topology generated by the basis of half-open intervals:

The topology is strictly finer than the Euclidean topology: every open interval $(a, b) = \bigcup_{n=1}^\infty [a + 1/n, b)$ is open in $\mathbb{R}_\ell$. Moreover, each $[a, b)$ is clopen (both open and closed)!

Properties of $\mathbb{R}_\ell$:

1. First-Countable: The countable family $\{[x, x + 1/n) : n \in \mathbb{N}\}$ forms a local basis at $x$.
2. Separable: The rational numbers $\mathbb{Q}$ are dense in $\mathbb{R}_\ell$ (every $[a, b)$ contains a rational).
3. Lindelöf: Every open cover of $\mathbb{R}_\ell$ has a countable subcover.
4. NOT Second-Countable: Any basis $\mathcal{B}$ for $\mathbb{R}_\ell$ must be uncountable! (Each point $x$ must be the unique left endpoint of some basis element).

2. The Sorgenfrey Plane $\mathbb{R}_\ell \times \mathbb{R}_\ell$

The product of two Lindelöf spaces is NOT necessarily Lindelöf!

Theorem 4.6 (Pathology of the Sorgenfrey Plane): The Sorgenfrey plane $\mathbb{S} = \mathbb{R}_\ell \times \mathbb{R}_\ell$ is:

1. First-countable and Separable ($\mathbb{Q} \times \mathbb{Q}$ is dense).
2. NOT Lindelöf!
3. NOT Normal ($T_4$)!

Proof that $\mathbb{R}_\ell^2$ is not Lindelöf: Consider the "anti-diagonal" line in $\mathbb{R}^2$:

For each point $p = (x, -x) \in L$, consider the basic open rectangle in $\mathbb{S}$:

Notice that for any other point $(y, -y) \in L$ with $y \ne x$: If $y > x$, then $-y < -x$, so $(y, -y) \notin U_x$ because its second coordinate is strictly less than $-x$. If $y < x$, then $y \notin [x, x+1)$. Thus:

The relative topology on $L$ is DISCRETE! Since $L$ is an uncountable discrete closed subspace, the open cover $\mathcal{U} = \{U_x : x \in \mathbb{R}\} \cup \{\mathbb{S} \setminus L\}$ of $\mathbb{S}$ contains no countable subcover. Therefore, $\mathbb{S} = \mathbb{R}_\ell \times \mathbb{R}_\ell$ is not Lindelöf. $\blacksquare$