First HW exercise is simply doing in reverse, with very little adjustements, the second part of the proof of the claim that 𐌱 = B(f, epsilon) is a basis on C([0,1]).
Awesome video! Why is the second property of base needed? Is it possible to build a set of elements that satisfies the first property (for all U open and x ∈ U there is a base set b such that x ∈ b ⊆ U) but not the second (for some x ∈ b1 ∩ b2 there is no b3 such that x ∈ b3 ⊆ b1 ∩ b2)?
@@r.maelstrom4810 thanks, I already figured it out and even left a comment under my question. Somehow it got removed by youtube. I misunderstood the definition, I thought that B is a subset of a topology T, but here instead it's defined as just a subset of P(X), and then a topology T is defined as an induced topology. If the definition is B some subset of P(X), then the property is needed. If the definition is B subset of T, then it works automatically by the first property of base plus properties of T
can you fix the playlist? the videos are in the wrong order
First HW exercise is simply doing in reverse, with very little adjustements, the second part of the proof of the claim that 𐌱 = B(f, epsilon) is a basis on C([0,1]).
Wonderful explanation ❤
The Axiom of Choice is strong in this one...
Awesome video!
Why is the second property of base needed? Is it possible to build a set of elements that satisfies the first property (for all U open and x ∈ U there is a base set b such that x ∈ b ⊆ U) but not the second (for some x ∈ b1 ∩ b2 there is no b3 such that x ∈ b3 ⊆ b1 ∩ b2)?
15:40
@@r.maelstrom4810 thanks, I already figured it out and even left a comment under my question. Somehow it got removed by youtube. I misunderstood the definition, I thought that B is a subset of a topology T, but here instead it's defined as just a subset of P(X), and then a topology T is defined as an induced topology. If the definition is B some subset of P(X), then the property is needed. If the definition is B subset of T, then it works automatically by the first property of base plus properties of T