In mathematics, a topological space is said to be limit point compact[1][2] or weakly countably compact[3] if every infinite subset of has a limit point in This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

Properties and examples

See also


  1. ^ The terminology "limit point compact" appears in a topology textbook by James Munkres where he says that historically such spaces had been called just "compact" and what we now call compact spaces were called "bicompact". There was then a shift in terminology with bicompact spaces being called just "compact" and no generally accepted name for the first concept, some calling it "Fréchet compactness", others the "Bolzano-Weierstrass property". He says he invented the term "limit point compact" to have something at least descriptive of the property. Munkres, p. 178-179.
  2. ^ Steen & Seebach, p. 19
  3. ^ Steen & Seebach, p. 19
  4. ^ Steen & Seebach, Example 6
  5. ^ Steen & Seebach, Example 50
  6. ^ Steen & Seebach, p. 20. What they call "normal" is T4 in wikipedia's terminology, but it's essentially the same proof as here.