Jump to content

Core-compact space

From Wikipedia, the free encyclopedia

In general topology and related branches of mathematics, a core-compact topological space is a topological space whose partially ordered set of open subsets is a continuous poset.[1] Equivalently, is core-compact if it is exponentiable in the category Top of topological spaces.[1][2][3] Expanding the definition of an exponential object, this means that for any , the set of continuous functions has a topology such that function application is a unique continuous function from to , which is given by the Compact-open topology and is the most general way to define it.[4]

Another equivalent concrete definition is that every neighborhood of a point contains a neighborhood of whose closure in is compact.[1] As a result, every (weakly) locally compact space is core-compact, and every Hausdorff (or more generally, sober[4]) core-compact space is locally compact, so the definition is a slight weakening of the definition of a locally compact space in the non-Hausdorff case.

See also

[edit]

References

[edit]
  1. ^ a b c "Core-compact space". Encyclopedia of mathematics.
  2. ^ Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. Vol. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0. MR 1975381. S2CID 118338851. Zbl 1088.06001.
  3. ^ Exponential law for spaces. at the nLab
  4. ^ a b Vladimir Sotirov. "The compact-open topology: what is it really?" (PDF).

Further reading

[edit]