structural:reformatting, stylistic:style_transfer
Closure operator and interior operator definitions

In terms of the interior operator, a function f: X → Y {\displaystyle f:X\to Y} between topological spaces is continuous if and only if for every subset B ⊆ Y, {\displaystyle B\subseteq Y,} f − 1 (int Y B) ⊆ int X (f − 1 (B)). {\displaystyle f^{-1}\left(\operatorname {int} _{Y}B\right)~\subseteq ~\operatorname {int} _{X}\left(f^{-1}(B)\right).}

In terms of the closure operator, f: X → Y {\displaystyle f:X\to Y} is continuous if and only if for every subset A ⊆ X, {\displaystyle A\subseteq X,} f (cl X A) ⊆ cl Y (f (A)). {\displaystyle f\left(\operatorname {cl} _{X}A\right)~\subseteq ~\operatorname {cl} _{Y}(f(A)).} That is to say, given any element x ∈ X {\displaystyle x\in X} that belongs to the closure of a subset A ⊆ X, {\displaystyle A\subseteq X,} f (x) {\displaystyle f(x)} necessarily belongs to the closure of f (A) {\displaystyle f(A)} in Y. {\displaystyle Y.} If we declare that a point x {\displaystyle x} is close to a subset A ⊆ X {\displaystyle A\subseteq X} if x ∈ cl X A, {\displaystyle x\in \operatorname {cl} _{X}A,} then this terminology allows for a plain English description of continuity: f {\displaystyle f} is continuous if and only if for every subset A ⊆ X, {\displaystyle A\subseteq X,} f {\displaystyle f} maps points that are close to A {\displaystyle A} to points that are close to f (A). {\displaystyle f(A).} Similarly, f {\displaystyle f} is continuous at a fixed given point x ∈ X {\displaystyle x\in X} if and only if whenever x {\displaystyle x} is close to a subset A ⊆ X, {\displaystyle A\subseteq X,} then f (x) {\displaystyle f(x)} is close to f (A). {\displaystyle f(A).}

Instead of specifying topological spaces by their open subsets, any topology on X {\displaystyle X} can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset A {\displaystyle A} of a topological space X {\displaystyle X} to its topological closure cl X A {\displaystyle \operatorname {cl} _{X}A} satisfies the Kuratowski closure axioms. Conversely, for any closure operator A ↦ cl A {\displaystyle A\mapsto \operatorname {cl} A} there exists a unique topology τ {\displaystyle \tau } on X {\displaystyle X} (specifically, τ:= { X ∖ cl A: A ⊆ X } {\displaystyle \tau :=\{X\setminus \operatorname {cl} A:A\subseteq X\}}) such that for every subset A ⊆ X, {\displaystyle A\subseteq X,} cl A {\displaystyle \operatorname {cl} A} is equal to the topological closure cl (X, τ) A {\displaystyle \operatorname {cl} _{(X,\tau)}A} of A {\displaystyle A} in (X, τ). {\displaystyle (X,\tau).} If the sets X {\displaystyle X} and Y {\displaystyle Y} are each associated with closure operators (both denoted by cl {\displaystyle \operatorname {cl} }) then a map f: X → Y {\displaystyle f:X\to Y} is continuous if and only if f (cl A) ⊆ cl (f (A)) {\displaystyle f(\operatorname {cl} A)\subseteq \operatorname {cl} (f(A))} for every subset A ⊆ X. {\displaystyle A\subseteq X.}

Similarly, the map that sends a subset A {\displaystyle A} of X {\displaystyle X} to its topological interior int X A {\displaystyle \operatorname {int} _{X}A} defines an interior operator. Conversely, any interior operator A ↦ int A {\displaystyle A\mapsto \operatorname {int} A} induces a unique topology τ {\displaystyle \tau } on X {\displaystyle X} (specifically, τ:= { int A: A ⊆ X } {\displaystyle \tau :=\{\operatorname {int} A:A\subseteq X\}}) such that for every A ⊆ X, {\displaystyle A\subseteq X,} int A {\displaystyle \operatorname {int} A} is equal to the topological interior int (X, τ) A {\displaystyle \operatorname {int} _{(X,\tau)}A} of A {\displaystyle A} in (X, τ). {\displaystyle (X,\tau).} If the sets X {\displaystyle X} and Y {\displaystyle Y} are each associated with interior operators (both denoted by int {\displaystyle \operatorname {int} }) then a map f: X → Y {\displaystyle f:X\to Y} is continuous if and only if f − 1 (int B) ⊆ int (f − 1 (B)) {\displaystyle f^{-1}(\operatorname {int} B)\subseteq \operatorname {int} \left(f^{-1}(B)\right)} for every subset B ⊆ Y. {\displaystyle B\subseteq Y.}