# Continuous Image of Compact Space is Compact/Corollary 2

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## 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): $5.5$: Continuous maps on compact spaces: Corollary $5.5.3$