Compactness and Continuous Functions

Topology · Axiom Academy

LESSON Compactness and Continuous Functions Understanding how continuous functions preserve compactness and create homeomorphisms The Fundamental Preservation Property One of the most powerful results in topology is that continuous functions preserve compactness. This property makes compactness a topological property and gives us powerful tools for proving results about continuous functions. Let be a continuous function and let be a compact subset of . Then is compact in . We prove this by taking an open cover of the image and pulling it back to the domain. The continuity of is crucial for making this work. Corollary: Compact Sets in Hausdorff Spaces are Closed Before we proceed to our main application, we need an important auxiliary result that connects compactness with the Hausdorff property. Let be a Hausdorff space. Then every compact subset of is closed. Continuous Bijections from Compact to Hausdorff Now we arrive at a remarkable theorem: a continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism. This means we get the continuity of the inverse "for free"! Let be a continuous bijection from a compact space to a Hausdorff space . Then is a homeomorphism. Proof: Why the Inverse is Continuous The key is to show that is continuous by proving it maps closed sets to closed sets. We use both the compactness of and the Hausdorff property of . Let's see how these theorems apply in concrete situations.

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