# finite neighborhood base

My question might seem obvious but still I want to sure if the result I shall state below is true:

if the neighborhood base in each point of a space say X is finite. Then we have a discrete topology.

Many thanks and apologies in case I just stated some insanity above :)

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## 2 Answers

No, this is not true. It would imply, for instance, that every finite topological space is discrete, but this is far from the case. For a trivial example, take the trivial (or indiscrete) topology on any finite set $X$ of cardinality greater than one. Or take the Sierpinski space, i.e., the unique two-point space (up to homeomorphism) with one open point and one closed point.
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