# Definition:Continuous Mapping (Topology)/Point/Definition 4

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## Definition

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.

Let $x \in S_1$.

The mapping $f$ is **continuous at (the point) $x$** (with respect to the topologies $\tau_1$ and $\tau_2$) if and only if:

- For every neighborhood $N$ of $\map f x$ in $T_2$, $f^{-1} \sqbrk N$ is a neighborhood of $x$.

## Also known as

If it is necessary to distinguish between multiple topologies on the same set, then the terminology **$\tuple {\tau_1, \tau_2}$-continuous** can be used to define a **continuous mapping**.