Definition:Continuous Mapping (Topology)/Point/Neighborhoods

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 $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continuous function (continuous mapping)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continuous function (continuous mapping, continuous map)