Chapter 12: Countability and Separation Axioms
In point‑set topology, countability axioms and separation axioms are two core tools for describing the degree of fineness of the structure of a topological space. Countability axioms (such as first countability, second countability, separability, and the Lindelöf property) restrict the cardinality of a “basis” or a “dense subset” of the topological space; separation axioms (such as \(T_0\), \(T_1\), etc.) describe to what extent points and points, or points and closed sets, can be separated by open sets. This chapter will explore these axioms in detail and their relationships.
1. First Countable and Second Countable Spaces
When studying the local properties of a topological space, we often hope that each point has a countable neighbourhood system nearby – this is the concept of first countability. If the entire space possesses a countable topological basis, it is called second countable.
Definition 12.1 (First Countable Space)
A topological space \((X, \mathcal{T})\) is called first countable if for every point \(x \in X\) there exists a countable open neighbourhood basis
satisfying: for any open set \(U\) containing \(x\), there exists some \(B_{x,i} \in \mathcal{B}_x\) such that
Definition 12.2 (Second Countable Space)
A topological space \((X, \mathcal{T})\) is called second countable if it possesses a countable basis
such that every open set in \(\mathcal{T}\) can be expressed as a union of elements of \(\mathcal{B}\) .
Basic relation: Clearly, second countability implies first countability. Indeed, if the space has a global countable basis \(\mathcal{B}\), then for any point \(x\), we simply select all basis elements that contain \(x\), i.e., \(\mathcal{B}_x = \{B_i \in \mathcal{B} : x \in B_i\}\), which automatically forms a countable local open neighbourhood basis at \(x\).
Example: Countable product space
Consider the countably infinite‑dimensional real space \(\mathbb{R}^\omega = \mathbb{R}^{\mathbb{N}}\) with the product topology \(\mathcal{T}_p\). A basis element has the form
where, except for finitely many \(i\), almost every interval is the whole \(\mathbb{R}\) (i.e., for almost all \(i\), \((a_i, b_i) = \mathbb{R}\)).
1.1 Hereditary Properties of Countability Axioms
Countability axioms behave well under taking subspaces and countable products.
Lemma 12.1 (Hereditary for subspaces)
If \((X, \mathcal{T})\) is first countable (or second countable), then any subspace \(A \subseteq X\) with the subspace topology is also first countable (or second countable).
Lemma 12.2 (Hereditary for countable products)
If for every \(i \in \mathbb{N}\) the space \(X_i\) is first countable (or second countable), then the countable product space \((\prod_{i \in \mathbb{N}} X_i, \mathcal{T}_p)\) is also first countable (or second countable).
Proof of Lemma 12.2 (for first countability) (Click to expand)
Assume each \(X_i\) is first countable. Take an arbitrary point in the product space:
Since each \(X_i\) is first countable, at each coordinate \(x_i\) there exists a countable open neighbourhood basis \(\mathcal{B}_{x_i}\).
Define a local basis at \(x\) in the product space by
Because each local basis \(\mathcal{B}_{x_i}\) is countable, and when forming the product we only allow finitely many coordinates to take basis elements other than the whole space, the set of such finite combinations is still countable.
Hence \(\mathcal{B}_x\) is a countable set. One can easily verify that \(\mathcal{B}_x\) indeed forms an open neighbourhood basis at \(x\) for the product topology. This proves that the product space is first countable. \(\square\)
2. Separable Spaces and Lindelöf Spaces
We now examine two other global topological properties related to countability: separability (restricting the cardinality of a dense set) and the Lindelöf property (restricting the size of open covers).
Definition 12.3 (Separable Space)
A topological space \((X, \mathcal{T})\) is called separable if it possesses a countable dense subset.
That is, there exists a countable set \(S = \{x_i\}_{i \in \mathbb{N}} \subseteq X\) such that its closure equals the whole space:
Definition 12.4 (Lindelöf Space)
A topological space \((X, \mathcal{T})\) is called a Lindelöf space if every open cover \(\mathcal{A} = \{A_i\}_{i \in I}\) of \(X\) has a countable subcover.
2.1 Powerful Consequences of Second Countability
Second countability is the strongest countability axiom; it directly implies separability and the Lindelöf property.
Proposition 12.3
If a topological space \((X, \mathcal{T})\) is second countable, then:
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(1) It is Lindelöf.
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(2) It is separable.
Proof of Proposition 12.3 (Click to expand)
Let \(\mathcal{B} = \{B_n\}_{n \in \mathbb{N}}\) be a countable topological basis of \(X\).
Proof of (1): Second countable \(\Rightarrow\) Lindelöf
Let \(\mathcal{A} = \{A_i\}_{i \in I}\) be an arbitrary open cover of \(X\). For each \(x \in X\), because \(\mathcal{A}\) covers \(X\), there exists some \(A_i \in \mathcal{A}\) with \(x \in A_i\).
Since \(\mathcal{B}\) is a basis, there exists some basis element \(B_{n_x} \in \mathcal{B}\) such that
Letting \(x\) range over \(X\), the selected basis elements form a subfamily \(\mathcal{B}'\) of \(\mathcal{B}\). Because \(\mathcal{B}\) itself is countable, \(\mathcal{B}'\) is countable (or finite).
For each basis element \(B_k \in \mathcal{B}'\), choose an arbitrary open set \(A_{i_k}\) from the original cover \(\mathcal{A}\) that contains it (i.e., \(B_k \subseteq A_{i_k}\)).
Set \(\mathcal{A}' = \{A_{i_k}\}\). Since \(\mathcal{B}'\) is countable, \(\mathcal{A}'\) is also countable. Because \(\mathcal{B}'\) covers \(X\) and each \(B_k \subseteq A_{i_k}\), \(\mathcal{A}'\) also covers \(X\). Thus we have found a countable subcover, proving the Lindelöf property.
Proof of (2): Second countable \(\Rightarrow\) Separable
From the countable basis \(\mathcal{B} = \{B_n\}_{n \in \mathbb{N}}\), for each non‑empty basis element \(B_n \ne \emptyset\), pick an arbitrary point \(x_n \in B_n\).
Let \(S = \{x_n\}_{n \in \mathbb{N}}\). Clearly \(S\) is a countable subset. We claim that \(\bar{S} = X\).
Using an equivalent characterisation of density: we need to show that for every non‑empty open set \(U \subseteq X\), \(U \cap S \ne \emptyset\).
Because \(\mathcal{B}\) is a basis, every non‑empty open set \(U\) contains some basis element \(B_k \in \mathcal{B}\). By the construction of \(S\), the point \(x_k\) belongs to \(B_k\), hence also to \(U\).
Moreover \(x_k \in S\), so \(x_k \in U \cap S\); therefore \(U \cap S \ne \emptyset\). This proves that \(S\) is dense in \(X\), so \(X\) is separable. \(\square\)
3. Relationships between Topological Properties and Classical Counterexamples
In metric spaces, many countability and compactness concepts are equivalent, but in general topological spaces their relationships form a one‑way implication structure.
Diagram of topological properties:
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In general topological spaces: Second countable \(\implies\) First countable Second countable \(\implies\) Separable Second countable \(\implies\) Lindelöf
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In metric spaces, the following are equivalent: Second countable \(\iff\) Separable \(\iff\) Lindelöf
(Note: In metric spaces, sequential compactness is also equivalent to compactness, and further connections exist under the condition of being closed and bounded.)
3.1 Classical Counterexample: The Lower Limit Topology \(\mathbb{R}_l\)
The lower limit topology \(\mathbb{R}_l\) on the real line is an extremely classical counterexample used to distinguish these topological properties. Its basis consists of half‑open intervals:
Properties of the Lower Limit Topology \(\mathbb{R}_l\)
- (1) It is first countable.
For any \(x \in \mathbb{R}\), we can construct a countable local neighbourhood basis:
It is easy to verify that any open set containing \(x\) must contain some element of \(\mathcal{B}_x\).
- (2) It is NOT second countable.
Suppose a basis \(\mathcal{B}'\) existed. For each real number \(x \in \mathbb{R}\), consider the open set \([x, x+1)\). Since \(\mathcal{B}'\) is a basis, there must exist some \(B_x \in \mathcal{B}'\) such that
This forces the infimum of \(B_x\) to be exactly \(x\), i.e., \(\inf B_x = x\).
If \(x \ne y\), then necessarily \(B_x \ne B_y\) (because their minima/infima are different). Thus we would need distinct basis elements for uncountably many distinct real numbers \(x\). Hence the cardinality of \(\mathcal{B}'\) would be at least \(\aleph_1\), i.e., the basis must be uncountable!
- (3) It is separable.
The set of rational numbers \(\mathbb{Q}\) is still dense in the lower limit topology, i.e., \(\bar{\mathbb{Q}} = \mathbb{R}\) (with respect to \(\mathcal{T}_l\)). Moreover \(\mathbb{Q}\) is countable.
- (4) It is Lindelöf.
(The proof is omitted here, but \(\mathbb{R}_l\) is indeed Lindelöf.)
Conclusion: \(\mathbb{R}_l\) is separable and Lindelöf, but it is not second countable. This breaks the equivalence of the three properties in metric spaces and also shows that \(\mathbb{R}_l\) cannot be metrizable.
4. Separation Axioms
Separation axioms investigate to what extent “point from point” and “point from closed set” can be separated by open sets in a topological space. From weakest to strongest, the main ones are:
Definition 12.5 (\(T_0\) Space)
A topological space \((X, \mathcal{T})\) is called a \(T_0\) space (Kolmogorov space) if for any two distinct points \(x \ne y\), at least one open set exists that contains one point but not the other.
That is, either there exists an open set \(U\) with \(x \in U\) and \(y \notin U\), or there exists an open set \(V\) with \(y \in V\) and \(x \notin V\).
Definition 12.6 (\(T_1\) Space)
A topological space \((X, \mathcal{T})\) is called a \(T_1\) space (Fréchet space) if for any two distinct points \(x \ne y\), each point has an open set that contains it but not the other.
That is, there exists an open set \(U\) with \(x \in U\) and \(y \notin U\), and there exists an open set \(V\) with \(y \in V\) and \(x \notin V\).
Difference between \(T_0\) and \(T_1\) and a counterexample: The \(T_1\) condition is strictly stronger than \(T_0\). There exist spaces that are \(T_0\) but not \(T_1\). A classic counterexample is the Sierpiński space: let \(X = \{a, b\}\) with topology \(\mathcal{T} = \{\emptyset, \{a\}, \{a, b\}\}\). For \(a \ne b\), the open set \(\{a\}\) contains \(a\) but not \(b\) (so \(T_0\) holds). However, there is no open set containing \(b\) but not \(a\) (so \(T_1\) fails).