Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

Let $X$ be a set and write $\mathcal{P}(X)$ for its power set. A topology on $X$ is a family $\mathcal{T} \subseteq \mathcal{P}(X)$ such that

• (T1) $\varnothing \in \mathcal{T}$ and $X \in \mathcal{T}$;
• (T2) $\bigcup \mathcal{S} \in \mathcal{T}$ for every $\mathcal{S} \subseteq \mathcal{T}$;
• (T3) $U \cap V \in \mathcal{T}$ for all $U, V \in \mathcal{T}$.

A topological space is a pair $(X, \mathcal{T})$ consisting of a set $X$ and a topology $\mathcal{T}$ on it; the elements of $X$ are its points. When only one topology is in play we write $X$ for the pair; when several are, the topology is always named.

The members of $\mathcal{T}$ are the open sets of $(X,\mathcal{T})$. A subset $F \subseteq X$ is closed when its complement $X \setminus F$ is open, and clopen when it is both open and closed. By (T1) both $\varnothing$ and $X$ are clopen. Open and closed are not opposites: a set may be neither, and it may be both, so "not open" is never a synonym for "closed".

Finite intersections, and the empty one. Axiom (T3) is stated for two sets because that is all that is ever needed: if $n \ge 1$ and $U_1, \dots, U_n \in \mathcal{T}$, then $U_1 \cap \dots \cap U_n \in \mathcal{T}$ follows by iterating (T3) $n-1$ times. The intersection of the empty family of subsets of $X$ is $X$, since the condition "lies in every member of the empty family" is vacuous; that case is covered by (T1) rather than by (T3), which is one of the two reasons $X \in \mathcal{T}$ is demanded outright. The other is that (T2) alone gives only $\bigcup \varnothing = \varnothing$.

The closed sets, and the dual axiomatisation. Put $\mathcal{C} := \{\, X \setminus U : U \in \mathcal{T} \,\}$, the family of closed sets. Then

• (C1) $\varnothing \in \mathcal{C}$ and $X \in \mathcal{C}$;
• (C2) $\bigcap \mathcal{D} \in \mathcal{C}$ for every nonempty $\mathcal{D} \subseteq \mathcal{C}$;
• (C3) $C \cup D \in \mathcal{C}$ for all $C, D \in \mathcal{C}$.

Conversely, if a family $\mathcal{C} \subseteq \mathcal{P}(X)$ satisfies (C1), (C2) and (C3), then $\mathcal{T} := \{\, X \setminus C : C \in \mathcal{C} \,\}$ is a topology on $X$ whose closed sets are exactly the members of $\mathcal{C}$. So a topology may be presented equally well by its open sets or by its closed sets, and this library uses whichever is shorter.

Both directions are one computation, and it is carried out here rather than assumed. The map $U \mapsto X \setminus U$ is an involution of $\mathcal{P}(X)$, hence a bijection, and it carries the two De Morgan identities

$X \setminus \bigcup_{i \in I} A_i = \bigcap_{i \in I} (X \setminus A_i) \quad (I \ne \varnothing), \qquad X \setminus \bigcap_{i \in I} A_i = \bigcup_{i \in I} (X \setminus A_i) \quad (I \ne \varnothing)$

term by term into each other. Given (T1)–(T3): $\varnothing = X \setminus X$ and $X = X \setminus \varnothing$ lie in $\mathcal{C}$, which is (C1); for nonempty $\mathcal{D} \subseteq \mathcal{C}$ the family $\mathcal{S} = \{\, X \setminus C : C \in \mathcal{D} \,\}$ lies in $\mathcal{T}$ and $\bigcap \mathcal{D} = X \setminus \bigcup \mathcal{S}$ is closed by (T2), which is (C2); and $C \cup D = X \setminus ((X \setminus C) \cap (X \setminus D))$ is closed by (T3), which is (C3). Given (C1)–(C3) the same three lines run backwards, the empty-union case $\bigcup \varnothing = \varnothing$ being supplied by (C1) rather than by (C2); and $\{\, X \setminus C : C \in \mathcal{C} \,\}$ has $\mathcal{C}$ as its family of complements because complementation is an involution.

Comparison of topologies. If $\mathcal{T}_1$ and $\mathcal{T}_2$ are topologies on the same set $X$ and $\mathcal{T}_1 \subseteq \mathcal{T}_2$, then $\mathcal{T}_1$ is coarser than $\mathcal{T}_2$ and $\mathcal{T}_2$ is finer than $\mathcal{T}_1$; the comparison is strict when the inclusion is. Inclusion is a partial order on the set of topologies on $X$ (Partial order and partially ordered set), and it is not a total one: two topologies on the same set may be incomparable.

• The topology is part of the data. Two different topologies on one set are two different topological spaces, exactly as two different metrics on one set are two different metric spaces. Every statement below that says "the space $X$" has a topology fixed in advance.

• Nothing is assumed beyond (T1)–(T3). In particular no separation property is built into the word space here: distinct points need not lie in disjoint open sets, singletons need not be closed, and both failures occur in the standard topologies defined on the next item. Where a separation property is used it is stated as a hypothesis.

• This library always writes coarser and finer. The synonyms smaller/larger are unambiguous, but weaker/stronger is used in both directions in the literature and is avoided here entirely.

• Why "arbitrary unions, finite intersections" and not the reverse. The asymmetry is not a convention: it is what makes the metric notion of open set (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) an instance, and an arbitrary intersection of open sets genuinely need not be open. The failure is recorded on this page as a false statement, with witnesses in two different spaces.