Is it true that openness is transitive?

Its clear for me in metric spaces. But how about topological spaces?

To make it clear:

Let $X$ be a topological space. Then it holds: If $U$ is open in $X$ and $V\subseteq U$ open in $U$ then $V$ is open in $X$. Of course we consider subspaces with the subset-topology.

Is this true? If yes. How it can be proven?

  • $\begingroup$ Indeed, it is a "if and only if". $\endgroup$ – PtF Jul 9 '17 at 22:52

If $V$ is open in $U$ then there exists an open subset $V_0$ in $X$ such that $V=U\cap V_0$. Since $V$ is the intersection of two open subsets in $X$, $V$ is open in $X$.

  • $\begingroup$ Oh. Quite easy. Thanks :) $\endgroup$ – Marm Jul 6 '14 at 12:39

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