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 [tex]Y[/tex] is a topological space, and [tex]X\subset Y[/tex], we can make the set [tex]X[/tex] to be a topological space by defining the open set for it as [tex]U\cap X[/tex], where [tex]U[/tex] is an open set of [tex]Y[/tex].

I would like to show that this indeed defines a topological space. But I failed to prove that there is the open set [tex]X[/tex] among those open sets defined above, i.e. [tex]U\cap X[/tex]. Anybody helps me?

Otherwise, we can easily see that
[tex] (U_1\cap X) \cap (U_2 \cap X) = (U_1\cap U_2)\cap X [/tex]
and
[tex] (U_1\cap X) \cup (U_2\cap X) = (U_1\cup U_2) \cap X [/tex]
and
[tex] \phi = \phi \cap X [/tex]
And I lack the final piece that the [tex]X[/tex] is contained in the collection of open sets of [tex]X[/tex] defined above.

Thanks!

George Jones

Let U be Y.

ismaili

aha! How stupid I was!
Just let [tex] U[/tex] be [tex] Y [/tex].
Thank you a lot.

micromass

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