Core-compact space: Difference between revisions
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Another equivalent defintition is that every neighbourhood of a point x contains a compact neighbourhood of x. As a result, every (weakly) [[locally compact]] space is core-compact, and every Hausdoff 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. |
Another equivalent defintition is that every neighbourhood of a point x contains a compact neighbourhood of x. As a result, every (weakly) [[locally compact]] space is core-compact, and every Hausdoff 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. |
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== References == |
== References == |
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Revision as of 10:38, 2 January 2023
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A core-compact space is a topological space whose partially ordered set of open subsets is a continuous poset. This is equivalent to the fact that is an Exponential object in the category Top of topological spaces.
Another equivalent defintition is that every neighbourhood of a point x contains a compact neighbourhood of x. As a result, every (weakly) locally compact space is core-compact, and every Hausdoff 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.
References
External links
Exponential law for spaces. at the nLab