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PRODUCTS OF PROTOPOLOGICAL GROUPS JULIE C. JONES
... the pullback topologies, we will be interested in saturated sets. In particular, we will say that a set U ⊆ G is saturated with respect to N ∈ ᏺ if for all x ∈ U , η−1 N (ηN (x)) ⊆ U. 2. Characterization and product theorems Theorem 2.1 (characterization theorem for protopological groups). Let (G, τ ...
... the pullback topologies, we will be interested in saturated sets. In particular, we will say that a set U ⊆ G is saturated with respect to N ∈ ᏺ if for all x ∈ U , η−1 N (ηN (x)) ⊆ U. 2. Characterization and product theorems Theorem 2.1 (characterization theorem for protopological groups). Let (G, τ ...
Problem Sheet 2 Solutions
... Suppose X is Hausdorff and let x, y ∈ X with x 6= y. Then there exist open sets U , V with x ∈ U , y ∈ V and U ∩ V = ∅. Since U and V are open and non-empty, we see that X \ U and X \ V are both finite and hence the union (X \ U ) ∪ (X \ V ) is finite. But (X \ U ) ∪ (X \ V ) = X \ (U ∩ V ) = X \ ∅ ...
... Suppose X is Hausdorff and let x, y ∈ X with x 6= y. Then there exist open sets U , V with x ∈ U , y ∈ V and U ∩ V = ∅. Since U and V are open and non-empty, we see that X \ U and X \ V are both finite and hence the union (X \ U ) ∪ (X \ V ) is finite. But (X \ U ) ∪ (X \ V ) = X \ (U ∩ V ) = X \ ∅ ...
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... Topology, in its most abstract sense, is the study of a family of subsets, called open sets, of some given set X, when subject to certain conditions based purely on set-theoretic operations. Namely, these conditions are that the intersection of two open sets is an open set, union of open sets is an ...
... Topology, in its most abstract sense, is the study of a family of subsets, called open sets, of some given set X, when subject to certain conditions based purely on set-theoretic operations. Namely, these conditions are that the intersection of two open sets is an open set, union of open sets is an ...
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... Let X be a topological space with T the topology defined on it. X is said to be uniformizable 1. there is a uniformity U defined on X, and 2. T = TU , the uniform topology induced by U. It can be shown that a topological space is uniformizable iff it is completely regular. Clearly, every pseudometri ...
... Let X be a topological space with T the topology defined on it. X is said to be uniformizable 1. there is a uniformity U defined on X, and 2. T = TU , the uniform topology induced by U. It can be shown that a topological space is uniformizable iff it is completely regular. Clearly, every pseudometri ...
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... Remark. It can be shown that B is isomorphic to the Boolean algebra of clopen sets in B ∗ . This is the famous Stone representation theorem. ...
... Remark. It can be shown that B is isomorphic to the Boolean algebra of clopen sets in B ∗ . This is the famous Stone representation theorem. ...
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... You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
Topology Ph.D. Qualifying Exam ffrey Martin Geo Mao-Pei Tsui
... (a) Show that A1 : S 1 7→ S 1 is homotopic to the identity map. (b) Show that Ak : S k 7→ S k is homotopic to the identity map if k is odd. 3. (a) The space G is a topological group meaning that G is a group and also a Hausdorff topological space such that the multiplication and map taking each elem ...
... (a) Show that A1 : S 1 7→ S 1 is homotopic to the identity map. (b) Show that Ak : S k 7→ S k is homotopic to the identity map if k is odd. 3. (a) The space G is a topological group meaning that G is a group and also a Hausdorff topological space such that the multiplication and map taking each elem ...
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... of X is defined as follows: • The objects of Π1 (X) are the points of X Obj(Π1 (X)) = X , • morphisms are homotopy classes of paths “rel endpoints” that is HomΠ1 (X) (x, y) = Paths(x, y)/ ∼ , where, ∼ denotes homotopy rel endpoints, and, • composition of morphisms is defined via concatenation of pat ...
... of X is defined as follows: • The objects of Π1 (X) are the points of X Obj(Π1 (X)) = X , • morphisms are homotopy classes of paths “rel endpoints” that is HomΠ1 (X) (x, y) = Paths(x, y)/ ∼ , where, ∼ denotes homotopy rel endpoints, and, • composition of morphisms is defined via concatenation of pat ...