Core-compact space

In general topology and related branches of mathematics, a core-compact topological space ${\displaystyle X}$ is a topological space whose partially ordered set of open subsets is a continuous poset.^[1] Equivalently, ${\displaystyle X}$ 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 ${\displaystyle Y}$, the set of continuous functions ${\displaystyle {\mathcal {C}}(X,Y)}$ has a topology such that function application is a unique continuous function from ${\displaystyle X\times {\mathcal {C}}(X,Y)}$ to ${\displaystyle Y}$, 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 ${\displaystyle U}$ of a point ${\displaystyle x}$ contains a neighborhood ${\displaystyle V}$ of ${\displaystyle x}$ whose closure in ${\displaystyle U}$ is compact.

• Locally compact space

References

Further reading

• "core-compact but not locally compact". Stack Exchange. June 20, 2016.

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