Let F x y and Let a Be a Subspace of X Prove That F a a y the Restriction of F to a is Continuous
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'[10 marks] Let X and Y be topological spaces Let A be & subspace of X and B subspace of Y. Prove that the product topology on Ax B is the same as the topology Ax B inherits a8 & subspace of X x Y.'
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Suppose X is a topological space having subsets Y and Z with Z €Y Let T be the topology OH where wC arC viewing Z as subspace of X and F the topology OH Z where We are viewing a5 subspace of Y. Prove T=f
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Video Transcript
Folks. So we need to prove the equality of two. Apologies. So let's go ahead and laugh. You be an open set in the first apology. T. Okay. So t. Is the topology on Z where we're viewing Z as a subspace of X. Right? So then there exists um the open in X. Such that you is equal to vi intersected with Z. Alright, This is the definition of being open in the substance. A subspace topology on Z. You are the intersection of an open setting X with Z. Okay. But then Z is contained and why? Right. Oops subset. So in fact we have you is also equal to the intersected so Z is contained and why? And in fact we have the V. Cap intersected with Y. This is open and wide by definition of the subspace topology on why. And finally, what is the last thing we need to note? We need to note that we can write you then as V. Intersected with Y. And then intersected with Z. Right. This works out because Z is just contained in why? So when we intersect with Y first, then with the We do indeed. Yet you still So. All right. So this directly implies that you is in this topology F. It's open in the subspace topology of what? So when we view you or when we view Z as a subspace of why? Um We're also open there. So on the other hand, if we let you be in F this direction should be more straightforward, I think. So then what does this mean? This means there exists vi open in. Why such that you is the intersected with Z. But then V. Open. And Y. This says that. So why we just have the subspace topology coming from X. Right. So V. Open and Y. This implies that what letter should I use this time there exists? Um Let's say the prime open in X. Such that V. Is this V prime intersected with Why? Okay. Let's think about what we're trying to show. We're trying to show that you is open in the subspace topology coming from X. So we want to show that U. Is equal to an open set in X. Intersected with Z. So then you we know this is V. Intersected with Z. But that is the prime intersected with Y intersected with Z. But now Z is contained inside Y. So this is actually just V. Prime intersected with Z. The smaller set contained in Y. Okay. And V. Prime is open in X. So now this implies that you is in fact for us. Let's see. So half. Yeah. So F. We started with um yep. Here we go. So you is therefore in T. So when we view Z as a subspace of X. It is open. Okay, so it gets a little confusing because you have to think about, okay, what's open and what's alright, It's how it all sounds a little confusing, but here the point is I found an open set the prime of X. So that its intersection with Z was equal to you. And I started with you being an intersection of an open set and why? Right. 15. But we've shown both equality. We've shown both contain mints now. So this, in fact implies at the set T is equal to F. And we're done.
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