First-countable implies compactly generated

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

The property of topological spaces of being first-countable is stronger than the property of being compactly generated.

Verbal statement

Any first-countable space is compactly generated.

Proof

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