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... Two topological spaces X and Y are Borel isomorphic if there is a Borel measurable function f : X → Y with Borel inverse. Such a function is said to be a Borel isomorphism. The following result classifies all Polish spaces up to Borel isomorphism. Theorem. Every uncountable Polish space is Borel iso ...
... Two topological spaces X and Y are Borel isomorphic if there is a Borel measurable function f : X → Y with Borel inverse. Such a function is said to be a Borel isomorphism. The following result classifies all Polish spaces up to Borel isomorphism. Theorem. Every uncountable Polish space is Borel iso ...
Math 8306, Algebraic Topology Homework 12 Due in-class on Wednesday, December 3
... = B × G as right G-spaces. 2. Let G and H be topological groups. Suppose P1 → B is a principal Gbundle and P2 → B is a principal H-bundle. Show P1 ×B P2 (pullback!) is a principal G × H-bundle. 3. Suppose G → H is a homomorphism of topological groups, and P → B is a principal G-bundle. Show that the ...
... = B × G as right G-spaces. 2. Let G and H be topological groups. Suppose P1 → B is a principal Gbundle and P2 → B is a principal H-bundle. Show P1 ×B P2 (pullback!) is a principal G × H-bundle. 3. Suppose G → H is a homomorphism of topological groups, and P → B is a principal G-bundle. Show that the ...
Appendix: Basic notions and results in general topology A.1
... Definition. Topological space is a pair (X, T ), where X is a set and T is a family of subsets of X, satisfying the following properties: (a) ∅ ∈ T , X ∈ T . S (b) If A ⊂ T is any subset, then A ∈ T . (c) For any two sets U, V ∈ T we have U ∩ V ∈ T . A family T with these properties is called a topo ...
... Definition. Topological space is a pair (X, T ), where X is a set and T is a family of subsets of X, satisfying the following properties: (a) ∅ ∈ T , X ∈ T . S (b) If A ⊂ T is any subset, then A ∈ T . (c) For any two sets U, V ∈ T we have U ∩ V ∈ T . A family T with these properties is called a topo ...
Quotient spaces
... (2) p: X → X/∼ should be a continuous map. Put more simply, we wish to topologize X/∼ in a way satisfying condition (2). There seems to be no good reason to place any further conditions on what a quotient space should be, so, with this motivation, we make the following definition. Definition. Suppos ...
... (2) p: X → X/∼ should be a continuous map. Put more simply, we wish to topologize X/∼ in a way satisfying condition (2). There seems to be no good reason to place any further conditions on what a quotient space should be, so, with this motivation, we make the following definition. Definition. Suppos ...
Chapter 3 Topological and Metric Spaces
... that the filter converges to a point x ∈ X, written F → x, if N(x) ⊂ F , i.e. F is finer than the filter of all neighbourhoods of the point x. It is equivalent to claim that τ(x) ⊂ F since the family τ(x) of all open neighbourhoods of the point x is a base of the filter N(x). We will say that a filt ...
... that the filter converges to a point x ∈ X, written F → x, if N(x) ⊂ F , i.e. F is finer than the filter of all neighbourhoods of the point x. It is equivalent to claim that τ(x) ⊂ F since the family τ(x) of all open neighbourhoods of the point x is a base of the filter N(x). We will say that a filt ...
Point-countable bases and quasi
... below) l) and a point-countable weak a-space is quasi-developable. From the latter result follows the result of Okuyama [13] that a collectionwise normal Tx c-space is metrizable iff it has a point-countable base and the result of Heath [9] that a Г 3 stratifiable space is metrizable iff it has a po ...
... below) l) and a point-countable weak a-space is quasi-developable. From the latter result follows the result of Okuyama [13] that a collectionwise normal Tx c-space is metrizable iff it has a point-countable base and the result of Heath [9] that a Г 3 stratifiable space is metrizable iff it has a po ...