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Topology

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Moderate

Weakest Condition For Continuity

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Let $X$ be a topological space with a topological basis ${\mathcal E}$. Let $f:X\to X$ be a map from $X$ to itself.

Which of the following is the WEAKEST condition SUFFICIENT for the continuity of $f$?

A

for every closed set $B\subset X$ the inverse image $f^{-1}(B)$ is also a closed set in $X$.

B

for every open set $A \subset X$ the inverse image $f^{-1}(A)$ is also an open set in $X$.

C

for every $E\in {\mathcal E}$ the inverse image $f^{-1}(E)$ is an open set in $X$.

D

for every $E\in {\mathcal E}$ the inverse image $f^{-1}(E)$ is also an element of ${\mathcal E}$.