Continuity from Union of Restrictions
Jump to navigation
Jump to search
Theorem
Let $T_1$ and $T_2$ be topological spaces.
Then $f: T_1 \to T_2$ is continuous if either:
- $\ds T_1 = \bigcup_{i \mathop = 1}^n V_i$ where:
- each $V_i$ is closed in $T_1$
- the restriction $f \restriction_{V_i}$ is continuous for each $V_i$
or:
- $\ds T_1 = \bigcup_{i \mathop \in I} U_i$ where:
- $I$ is any (possibly infinite) indexing set
- each $U_i$ is open in $T_1$
- the restriction $f \restriction_{U_i}$ is continuous for each $U_i$.
Proof
First assertion
Let $\ds T_1 = \bigcup_{i \mathop = 1}^n V_i$, with the $V_i$ closed in $T_1$.
Assume that for each $i$, the restriction $f \restriction_{V_i}$ is continuous.
Then $f$ satisfies the hypotheses of Continuous Mapping on Finite Union of Closed Sets.
Hence $f$ is continuous.
$\blacksquare$
Second assertion
Let $\ds T_1 = \bigcup_{i \mathop \in I} U_i$, with the $U_i$ open in $T_1$.
Assume that for each $i \in I$, the restriction $f \restriction_{U_i}$ is continuous.
Then $f$ satisfies the hypotheses of Continuous Mapping on Union of Open Sets.
Hence $f$ is continuous.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 33$