Covering spaces
... and consider the path space P(X, x) ' hom((I, 0), (X, x)). Note that the path space is a pointed space (with basepoint given by the constant path) and is contractible. Let X̃ be the quotient of P(X, x) obtained by identifying paths that are path-homotopic and let x̃ = [cx ]. There is a natural map p ...
... and consider the path space P(X, x) ' hom((I, 0), (X, x)). Note that the path space is a pointed space (with basepoint given by the constant path) and is contractible. Let X̃ be the quotient of P(X, x) obtained by identifying paths that are path-homotopic and let x̃ = [cx ]. There is a natural map p ...
... Definition 2.21: A topological space X is said to be λ - Lindelof if every λ -open cover of X has a countable subcover. A subset A of a topological space X is said to be λ - Lindelof relative to X if every cover of A by λ -open sets of X has a countable subcover. Theorem 2.22: Every λ - Lindelolf sp ...
Compactly Generated Domain Theory
... Theorem 3.11 The category kP is a full reflective exponential ideal of kTop. It follows that kP is complete and cocomplete. Limits are calculated as in kTop. Colimits are calculated by reflecting colimits from kTop. Thus, in kP, neither limits nor colimits are, in general, calculated as in Top (thou ...
... Theorem 3.11 The category kP is a full reflective exponential ideal of kTop. It follows that kP is complete and cocomplete. Limits are calculated as in kTop. Colimits are calculated by reflecting colimits from kTop. Thus, in kP, neither limits nor colimits are, in general, calculated as in Top (thou ...
Full Text Article - International Journal of Mathematics
... intersections (resp. Finite unions). The δ–closure of A [25] is the set of all x in x such that the interior of every closed neighbourhood of x intersects A non trivally. The δ–closure of A is denoted by clδ (A) or δcl(A) . The δ–interior of a subset A of X is the union of all regular open sets of X ...
... intersections (resp. Finite unions). The δ–closure of A [25] is the set of all x in x such that the interior of every closed neighbourhood of x intersects A non trivally. The δ–closure of A is denoted by clδ (A) or δcl(A) . The δ–interior of a subset A of X is the union of all regular open sets of X ...
Abelian Categories
... objects, and a single morphism between two open sets U and V if and only if U ⊂ V. The empty set is the initial object of this category, and X is the terminal object. This is a special case of the case "partially ordered set", mentioned above. Take P:= the set of open subsets If X is a topological s ...
... objects, and a single morphism between two open sets U and V if and only if U ⊂ V. The empty set is the initial object of this category, and X is the terminal object. This is a special case of the case "partially ordered set", mentioned above. Take P:= the set of open subsets If X is a topological s ...
Notes on Topology
... such that p is the only member of E which belongs to it: N ∩ E = { p }. Exercise 27 Show that p ∈ E is not isolated if and only if p ∈ E \ { p} or, equivalently, if there exists a net in E \ { p} which converges to p. Exercise 28 Let p ∈ X . Show that p ∈ E \ { p} if and only if the family of subset ...
... such that p is the only member of E which belongs to it: N ∩ E = { p }. Exercise 27 Show that p ∈ E is not isolated if and only if p ∈ E \ { p} or, equivalently, if there exists a net in E \ { p} which converges to p. Exercise 28 Let p ∈ X . Show that p ∈ E \ { p} if and only if the family of subset ...
INTRODUCTION TO ALGEBRAIC TOPOLOGY 1.1. Topological
... ⊲ morphisms: continuous maps, equipped with composition of continuous maps. Explicitly HomTop (X, Y ) is the set of continuous maps from X to Y and composition is a set map ◦ : HomTop (Y, Z) × HomTop (X, Y ) → HomTop (X, Z). These satisfy the Axioms of a category: the existence and properties of ide ...
... ⊲ morphisms: continuous maps, equipped with composition of continuous maps. Explicitly HomTop (X, Y ) is the set of continuous maps from X to Y and composition is a set map ◦ : HomTop (Y, Z) × HomTop (X, Y ) → HomTop (X, Z). These satisfy the Axioms of a category: the existence and properties of ide ...
Paracompact subsets
... In this paper, we distinguish 3 types of paracompact subsets and 2 types of countably paracompact subsets. Definition 1. A subset M of a topological space (X, 2T) is a-paracompact (a-paracompact) if every open cover by members of ST has an open locally finite (d-locally finite) refinement by members ...
... In this paper, we distinguish 3 types of paracompact subsets and 2 types of countably paracompact subsets. Definition 1. A subset M of a topological space (X, 2T) is a-paracompact (a-paracompact) if every open cover by members of ST has an open locally finite (d-locally finite) refinement by members ...
Two new type of irresolute functions via b-open sets
... ∩ g −1 (U2 ). Then U is a regular open set containing x and U ∩ A 6= ∅. Hence we have x ∈ / Clδ (A). This completes the proof. (ii). Follows from (i). ...
... ∩ g −1 (U2 ). Then U is a regular open set containing x and U ∩ A 6= ∅. Hence we have x ∈ / Clδ (A). This completes the proof. (ii). Follows from (i). ...
The homotopy category is a homotopy category. Arne Str¢m
... (with the obvious base point) is also well-pointed. the fiber of ...
... (with the obvious base point) is also well-pointed. the fiber of ...
seminar notes - Andrew.cmu.edu
... Lindelöf (indeed, it is compact; X ∗ is the one-point compactification of the discrete topology on R), but it is not separable since all of the sets {x} remain open for x 6= ∗. ...
... Lindelöf (indeed, it is compact; X ∗ is the one-point compactification of the discrete topology on R), but it is not separable since all of the sets {x} remain open for x 6= ∗. ...
on generalized closed sets
... Closed sets are fundamental objects in a topological space. For example, one can define the topology on a set by using either the axioms for the closed sets or the Kuratowski closure axioms. In 1970, N. Levine [16] initiated the study of so-called generalized closed sets. By definition, a subset S o ...
... Closed sets are fundamental objects in a topological space. For example, one can define the topology on a set by using either the axioms for the closed sets or the Kuratowski closure axioms. In 1970, N. Levine [16] initiated the study of so-called generalized closed sets. By definition, a subset S o ...