How to show induced topological space

[ismaili]

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!

 

[George Jones]

Let U be Y.

 

[ismaili]

Let U be Y.

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

 

[micromass]

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