Hausdorff implies KC

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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Statement

Property-theoretic statement

Here are two equivalent formulations:

• The property of being a Hausdorff space is stronger than the property of being a KC-space
• The property of being a compact space is stronger than the property of being a US-space

Verbal statement

Any compact subset of a Hausdorff space is closed.

Proof

Suppose $X$ is a Hausdorff space and $A$ is a closed subset of $X$ that is compact in the subspace topology. We need to show that $A$ is closed in $X$. In order to do this, it suffices to show that the complement of $A$ in $X$ is an open subset. In fact, it suffices to show that for any $x\in A\setminus X$, there is an open subset $U$ of $X$ containing $x$, and disjoint from $A$.

Let's do this:

• For every $y\in A$, use the Hausdorffness of $X$ to find disjoint open subsets $U_y\ni x$ and $V_y\ni y$
• Now the $V_y$s, as $y\in A$, cover $A$, hence $V_y\cap A$ form an open cover of $A$ in the subspace topology. By compactness of $A$, we can find a finite collection $y_1,y_2,\dots ,y_n\in A$ such that the union of $V_{y_1},V_{y_2},\dots ,V_{y_n}$ contains $A$
• Now consider the intersection:

$U=\bigcap _{i=1}^n{U}_{y_i}$

This is an open set containing $x$, because it is a finite intersection of open sets, and it is disjoint from each of the $V_{y_i}$s, hence it is disjoint from $A$. This completes the proof.

Note that the crucial thing is to pass to a finite subcover of $A$, so that we need to intersect only finitely many open subsets containing $x$, because the axioms for open subsets only guarantee that finite intersections of open subsets are open.

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 165 (Theorem 26.3) Also see Lemma 26.4, that provides a slightly more general formulation useful in other proofs, for instance, compact Hausdorff implies normal
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 27