Suppose we have a map $ f: X \rightarrow Y $ where $ X $ and $ Y $ are topological spaces. Also suppose that $ U $ and $ V $ are open subsets of $ X. $
Suppose that $ V \supset U $ if and only if $ f(V) \supset f(U). $ Is this another way of saying that $ f $ is a homeomorphism? If so, why?
Define $f$ to be the identity map from $\mathbb{R}$ with the standard topology to $\mathbb{R}$ with the discrete topology. Then for any two sets (even if they are not open) $U,V$ we have $U\subset V$ if and only if $f(U)\subset f(V)$. But obviously $f$ is not a homeomorphism. It is not even continuous.