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!