How Do Subspace Topologies Compare When One is Finer?

[tomboi03]

If T and T' are topologies on X and T' is strictly finer than T, what can you say about the corresponding subspace topologies on the subset Y of X?

See the thing is... I know that they CAN be finer but that's might not be for every case. because Y is JUST a subset of X. This does not necessarily mean finer or coarser.

How do i put this in terms of symbols in my proof?

 

[zhentil]

I would reread the definition of subspace topology.

For your question, suppose A is open in the subspace topology induced by T. Then by definition, there exists U open in X such that A = U int Y. Since U is also open in the T' topology this shows that A is open in the topology induced by T'. This shows that the T' induced topology is finer than the T-induced topology.

But it needn't be strictly finer (Hint: let Y be a single point).