Proving Equality of Subspace Topologies: A Topological Lemma Approach

[MathematicalPhysicist]

Well it's not homewrok cause i don't need to hand this question in, this is why i decided to put it here. (that, and there isn't a topology forum per se, perhaps it's suited to point set topology so the set theory forum may suit it).

Now to the question:
Show that if Y is a subspace of X, and A is a subset of Y, then the subspace topoology on A as a subspace of Y is the same as the subspace topology on A as a subspace of X.
(^-stands for intersection).
well so i wrote the specific topologies in hand:
T is a topology on X and T' a topology on Y.
T_Y subspace topology on Y from T, i.e: $$T_Y$$={ Y^U|U\in T }
$$T_{A_Y}$$={ A^U|U\in T' } is the subspace topology on A as a subspace of Y, and $$T_{A_X}$$={ A^U| U\in T }
is the subspace on A as a subspace of X, now i need to prove that the latter topologies are the same, i.e are equal.
and so iv'e begun:
let V be in T_{A_Y} then there exists U in T' such that V=A^U but A is a subset of Y, then V is a subset of Y^U, so because Y and U are in T', also Y^U in T', now i need to prove that U is in T (the topology of X), now there's a lemma that ithink i need to use here but not certain how to apply it, the lemma states that:
Let Y be a subspace of X, if U is open in Y and Y is open in X then U is open in X.
now i obvioulsy want to show that U is open in X, so i need Y to be open in X, not sure how to do it, any hints?
p.s
also hints for the second part of the proof would be nice but i haven't yet tried to prove the second part.

 

[ircdan]

Show that if Y is a subspace of X, and A is a subset of Y, then the subspace topoology on A as a subspace of Y is the same as the subspace topology on A as a subspace of X.

Suppose O is open in the topology on A induced from Y. Then 0 = A n V for some V open in Y. Since V is open Y we have V = Y n B for some B open in X. Then 0 = A n V = A n Y n B = A n B because A is a subset of Y , hence O is open in the topology induced on A from X.

Conversely, suppose O is open in the subspace topology on A induced from X. Then O = A n B for some open B in X. Since A is a subset of Y, O = A n B = A n (Y n B) and Y n B is open in Y since B is open in X. Thus O is open in the topology induced on A from Y.

So the open sets in the topology induced on A from Y are exactly the same as the open sets in the topology induced on A from X. So these topologies coincide.

Notice two things. I didn't use any theorems, only what it means to be open in each respective topology, and, I didn't introduce any symbols for the topologies, which makes things easier and cleaner imho. Hope this helps.

 

[MathematicalPhysicist]

I don't quite understand.
why because V is open in Y we have V=YnB for some B open in X, the fact that it's open in the subspace topology means that it's open in every other topology on Y, cause i think that this is what is assumed here, cause the above assertion is valid when V is in T_Y although from our assumption it's in T'.

 

[morphism]

U is open in the subspace A of Y
iff U=A$\cap$V where V is open in the subspace Y of X
iff U=A$\cap$(Y$\cap$G)=A$\cap$G where G is open in X (the second equality holds because A$\cap$Y=A).

 

[MathematicalPhysicist]

OK morphism repeating it doesn't explain it.

as I said U is in T_A_Y iff there exists V in T' such that U=AnV where T' is the topology on Y, the definition of subspace topology is quite clear every intersection of an open set in the topology on Y with A is in this topology.
now, if V is in T' does it means that it's in T_Y?
I mean does a set has only one topology on it ( i think not).

 

[morphism]

It's stated that the topology on Y is the subspace topology it gets from X...

 

[MathematicalPhysicist]

you mean that by saying Y is a subspace of X, it's actually saying that the subspace topology is the topology on Y.
yes, I see your point, thanks.