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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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up vote 5 down vote accepted

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.

Thinking about these examples may help you formulate further conditions on your space sufficient for what you want to hold. Hint: think about separation axioms.

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Good answer, I hope you didn't mind about the typo I corrected. –  Asaf Karagila May 13 '11 at 11:38
    
@Asaf: no, of course not. Thanks for doing so. –  Pete L. Clark May 13 '11 at 11:40
    
Thanks for the examples and the hint –  El Moro May 13 '11 at 11:55
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Am I misreading this question? It seems like it's obviously not true, for example any finite non-discrete space.

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thanks hehehe well that was a rough thought in my mind and I stated it the way it seemed to me –  El Moro May 13 '11 at 11:56
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