# Math Help - closure

1. ## closure

how

2. The conclusion is not true if f is not continous function.
If the contious condition is assumed, then it follows immediately from the continuousity of continous function.

3. I am sorry

The function f is continuous F:X tends to Y
and X and Y are two complete metric spaces

Thank you

4. Suppose that V is a subset. By the continuity of f we have $f^{-1}(\overline{f(V)})$ is closed in X (f^-1 from Y to X maps closed/open sets to closed/open sets resp.).
Notice that $f(V) \subset \overline{f(V)} \implies V\subset f^{-1}(\overline{f(V)})$. As the closure is the smallest closed superset of V we have $\overline{V}\subset f^{-1}(\overline{f(V)})$. Now take f of both sides to get the result.

You don't need completeness (or the fact that it is a metric space).

5. ........