tags:definition |topology |functional

Topological Continuous Function

Context

Statements

Definition (@Tu11 pp. 327, @Bou95, Chapter 2, Definition 1). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces. A mapping $f:X\to Y$ is said to be continuous at a point $x\in X$ if, for every topological neighborhood $N_Y$ of $f(x)\subset Y$, there exists a topological neighborhood $N_X$ of $x\in X$ such that the relation $x\in N_X$ implies $f(x)\in N_Y$. In addition, $f$ is continuous on $X$ if it is continuous at every point of $X$.