The Unapologetic Mathematician

Mathematics for the interested outsider

The Image of a Compact Space

One of the nice things about connectedness is that it’s preserved under continuous maps. It turns out that compactness is the same way — the image of a compact space X under a continuous map f:X\rightarrow Y is compact.

Let’s take an open cover \{U_i\} of the image f(X). Since f is continuous, we can take the preimage of each of these open sets \{f^{-1}(U_i)\} to get a bunch of open sets in X. Clearly every point of X is the preimage of some point of f(X), so the f^{-1}(U_i) form an open cover of X. Then we can take a finite subcover by compactness of X, picking out some finite collection of indices. Then looking back at the U_i corresponding to these indices (instead of their preimages) we get a finite subcover of f(X). Thus any open cover of the image has a finite subcover, and the image is compact.

January 16, 2008 - Posted by John Armstrong | Point-Set Topology, Topology | | 3 Comments

3 Comments »

  1. I’ve been lurking and reading your topology posts. Please keep it up!

    Comment by Tony | January 16, 2008

  2. iam in seargh on a filters and topology have you any information?

    Comment by bushi | February 4, 2008

  3. [...] subset of the disk of radius . Now the function is a continuous, real-valued function on , and the image of a compact space is compact, so takes some maximum value on [...]

    Pingback by Uniform Convergence of Power Series « The Unapologetic Mathematician | September 10, 2008

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