Compact implies feebly compact

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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 (i.e., compact space) must also satisfy the second topological space property (i.e., feebly compact space)
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Statement

Any compact space is a feebly compact space.

Definitions used

Compact space

Further information: Compact space

A topological space is termed compact if every open cover of the space has a finite subcover.

Feebly compact space

A topological space is termed feebly compact if every locally finite collection of nonempty open subsets is finite.

Proof

Given: A compact space X, a locally finite collection of nonempty open subsets V_i, i \in I of X.

To prove: I is finite.

Proof:

  1. There exists a point-indexed open cover U_x, x \in X of X such that each U_x intersects finitely many V_is: For each point x \in X, we can find an open subset U_x that works by the definition of locally finite. Putting these together, there exists a point-indexed open cover.
  2. There exists a finite subset A of X such that the U_a, a \in A, cover X: This follows from the previous step and the definition of compactness.
  3. The size of I is finite: Every V_i,i \in I, is a nonempty open subset of X, hence it intersects at least one of the U_as. Thus, to count the V_is, it suffices to count the V_is that intersect at least one U_a. For each U_a, the number of V_is intersecting it is finite, and there are finitely many as in A, so the total number is finite.