Definition:Continuous Mapping (Topology)/Point/Filters

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.

Also see

• Equivalence of Definitions of Continuous Mapping between Topological Spaces at Point