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... Call a set X with a closure operator defined on it a closure space. Every topological space is a closure space, if we define the closure operator of the space as a function that takes any subset to its closure. The converse is also true: Proposition 1. Let X be a closure space with c the associated ...
... Call a set X with a closure operator defined on it a closure space. Every topological space is a closure space, if we define the closure operator of the space as a function that takes any subset to its closure. The converse is also true: Proposition 1. Let X be a closure space with c the associated ...
Homology and cohomology theories on manifolds
... spaces to triples. For our purposes, a triple (X; X1 , X2 ) consists of a topological space X and two subspaces X1 and X2 with X = X1 ∪ X2 . Of course, a map of triples f : (X; X1 , X2 ) → (Y ; Y1 , Y2 ) is a continuous map f : X → Y with f (Xi ) ⊆ Yi . We denote by f1 : X1 → Y1 , f2 : X2 → Y2 and f ...
... spaces to triples. For our purposes, a triple (X; X1 , X2 ) consists of a topological space X and two subspaces X1 and X2 with X = X1 ∪ X2 . Of course, a map of triples f : (X; X1 , X2 ) → (Y ; Y1 , Y2 ) is a continuous map f : X → Y with f (Xi ) ⊆ Yi . We denote by f1 : X1 → Y1 , f2 : X2 → Y2 and f ...
Lecture 4
... Example 4.1: (i) Suppose X is a compact space and Y is a Hausdorff space then any surjective continuous map f : X −→ Y is a closed map. (ii) The reader may check that φ : R −→ S 1 given by φ(t) = exp(2πit) is an open mapping. (iii) The map φ : [0, 1] −→ S 1 given by φ(t) = exp(2πit) is closed but n ...
... Example 4.1: (i) Suppose X is a compact space and Y is a Hausdorff space then any surjective continuous map f : X −→ Y is a closed map. (ii) The reader may check that φ : R −→ S 1 given by φ(t) = exp(2πit) is an open mapping. (iii) The map φ : [0, 1] −→ S 1 given by φ(t) = exp(2πit) is closed but n ...