# How to show induced topological space

by ismaili
Tags: induced, space, topological
 P: 161 I am beginning to read about the topology, I met a problem puzzled me for a while. If $$Y$$ is a topological space, and $$X\subset Y$$, we can make the set $$X$$ to be a topological space by defining the open set for it as $$U\cap X$$, where $$U$$ is an open set of $$Y$$. I would like to show that this indeed defines a topological space. But I failed to prove that there is the open set $$X$$ among those open sets defined above, i.e. $$U\cap X$$. Anybody helps me? Otherwise, we can easily see that $$(U_1\cap X) \cap (U_2 \cap X) = (U_1\cap U_2)\cap X$$ and $$(U_1\cap X) \cup (U_2\cap X) = (U_1\cup U_2) \cap X$$ and $$\phi = \phi \cap X$$ And I lack the final piece that the $$X$$ is contained in the collection of open sets of $$X$$ defined above. Thanks!
 Mentor P: 5,916 Let U be Y.
P: 161
 Quote by George Jones Let U be Y.
aha! How stupid I was!
Just let $$U$$ be $$Y$$.
Thank you a lot.

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