Continuous Image of Compact Space is Compact/Corollary 2

Corollary to Continuous Image of Compact Space is Compact

A continuous mapping from a compact topological space to a metric space is bounded.

Proof

Follows from Continuous Image of Compact Space is Compact and Compact Subspace of Metric Space is Bounded.

$\blacksquare$

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $4.15$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.5$: Continuous maps on compact spaces: Corollary $5.5.3$