Continuous Images of Compact Sets

Topology · Axiom Academy

EXAMPLE Continuous Images of Compact Sets Three worked examples demonstrating that continuous images preserve compactness Let be a continuous function from a topological space to a topological space . If is a compact subset of , then is a compact subset of . The Closed Unit Interval Under Squaring Wrapping the Unit Interval Around a Circle These three examples demonstrate the power of the theorem that continuous images of compact sets are compact: Example 1: A compact interval mapped by a polynomial remains compact Example 2: A compact interval can be wrapped into a circle, preserving compactness Example 3: A compact 2D set projects to a compact 1D set Based on the example above, which approach is correct?

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