Introduction to Topology

Functions

Statements

Definition (@Men90, Definition 3.2). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A mapping $f:X\to Y$ is said to be continuous at $a\in X$ if, for every $\epsilon >0$, there exists $\delta >0$, such that the inequality $d_Y(f(x),f(a))<\epsilon$ implies $d_X(x,a)<\delta$.

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Continuity

Statements

Theorem (@Men90, Theorem 6.3). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, the function $f:(X,d_X)\to (Y,d_Y)$ is a metrical continuous function if, and only, if it is a topological continuous function.

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