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I can't see how the following is proved.

Given two topological space (X,T), (Y,U) and a functionffrom X to Y and the following two statements.

1.fis continuous, i.e. for every open set U inU, the inverse imagef(U) is in^{-1}T

2. For every convergent filter baseF->x, the induced filter basef[[F]] ->f(x)

it is claimed that statement 1 and statement 2 are equivalent, i.e. 1 if and only if 2

I can prove 1 -> 2

but how do i prove 2 -> 1

thanks a lot !

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# Convergent Filter Base and Continuous Function

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