# How to show induced topological space

1. May 30, 2011

### ismaili

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!

2. May 31, 2011

### George Jones

Staff Emeritus
Let U be Y.

3. May 31, 2011

### ismaili

aha! How stupid I was!
Just let $$U$$ be $$Y$$.
Thank you a lot. :shy:

4. May 31, 2011

### micromass

Staff Emeritus
Also, you might want to use [ itex] instead of [ tex ] if you don't want to start a new line every time...