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

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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) $x$** if and only if:

- for any filter $\FF$ on $T_1$ that converges to $x$, the corresponding image filter $f \sqbrk \FF$ converges to $\map f x$.

## 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**.