Topology question - Compact subset on the relative topology

[Pyroadept]

Homework Statement

Let (X,Ʈ) be a topological space and T \subseteq X a compact subset.
Show that T is compact as a subset of the space (T,Ʈ_T) where Ʈ_T is the relative topology on T.

Homework Equations

The Attempt at a Solution

Hi everyone,

Here's what I've done so far:

T is compact means that each open cover of T has a finite subcover.

The relative topology Ʈ_T = {T \cap V: V \in Ʈ}

Take an open cover of T over the topological space (T, Ʈ_T)

We need to show that this open cover has a finite subcover.

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Am I going in the right direction? Have I set up the answer correctly so far?
I think I need to do something involving intersections (as in the definition for the relative topology) but I'm not sure how to go about this.
Would someone be able to please give me a hint as to how to do this? Or is this completely wrong?

Thanks for any help!

 

[micromass]

So far you're on the right way.

What are the open sets in the subspace topology? Can you apply this to your open cover?