the structure of locally connected topological spaces
... by one of the authors (Youngs [9]). Moore employed two primitive concepts: region and contiguity (compare this with satelliticity 4.8). Youngs used the notion of arc as primitive. In the following pages a solution is given which is based upon the usual concept of open set. 0.2. The work is divided i ...
... by one of the authors (Youngs [9]). Moore employed two primitive concepts: region and contiguity (compare this with satelliticity 4.8). Youngs used the notion of arc as primitive. In the following pages a solution is given which is based upon the usual concept of open set. 0.2. The work is divided i ...
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... points of a space, introduce a space having a uniform base at nonisolated points and describe it as an image of a metric space by open boundary-compact maps. Some relationships among the images of metric spaces under open boundary-compact maps, pseudo-open boundary-compact maps, open compact maps, a ...
... points of a space, introduce a space having a uniform base at nonisolated points and describe it as an image of a metric space by open boundary-compact maps. Some relationships among the images of metric spaces under open boundary-compact maps, pseudo-open boundary-compact maps, open compact maps, a ...
Some Remarks on Semi Open Sets with Respect to an Ideal
... Theorem 5. Let X be a topological space and I an ideal such that the collection of open sets satisfies the finite intersection property, if A and B are weakly I-semi open, then so is A ∩ B. Proof. Since A and B are weakly I-semi open sets, there exist open sets U, V such that U \ A ∈ I and V \ B ∈ I ...
... Theorem 5. Let X be a topological space and I an ideal such that the collection of open sets satisfies the finite intersection property, if A and B are weakly I-semi open, then so is A ∩ B. Proof. Since A and B are weakly I-semi open sets, there exist open sets U, V such that U \ A ∈ I and V \ B ∈ I ...
Unitary Group Actions and Hilbertian Polish
... Let H be an infinite-dimensional separable complex Hilbert space and let U (H) be the group of all unitary tranformations of H. When U (H) is endowed with the strong operator topology it becomes a Polish group, and we denote it by U ∞ (following [7]). A more standard notation for this topological gr ...
... Let H be an infinite-dimensional separable complex Hilbert space and let U (H) be the group of all unitary tranformations of H. When U (H) is endowed with the strong operator topology it becomes a Polish group, and we denote it by U ∞ (following [7]). A more standard notation for this topological gr ...
The Closed Limit Point Compactness
... = , ∅, { , }, { , } . The sets = { , } and = { , } are closed in and has limit point i.e. = and = . Hence is a closed limit point compact. (ii) The real numbers with the usual topology R , is not a closed limit point compact. {3} is non-empty proper closed set in R which has no a limit point compact ...
... = , ∅, { , }, { , } . The sets = { , } and = { , } are closed in and has limit point i.e. = and = . Hence is a closed limit point compact. (ii) The real numbers with the usual topology R , is not a closed limit point compact. {3} is non-empty proper closed set in R which has no a limit point compact ...
Manifolds and Topology MAT3024 2011/2012 Prof. H. Bruin
... An equivalence relation ∼ on a space X is a relation on a set which is 1. Reflexive: x ∼ x. 2. Symmetric: x ∼ y if and only if y ∼ x. 3. Transitive: x ∼ y and y ∼ z imply x ∼ z. The set {y ∈ X : y ∼ x} is the equivalence class of x; it is denoted as [x]. Definition 20 Given an equivalence relation ∼ ...
... An equivalence relation ∼ on a space X is a relation on a set which is 1. Reflexive: x ∼ x. 2. Symmetric: x ∼ y if and only if y ∼ x. 3. Transitive: x ∼ y and y ∼ z imply x ∼ z. The set {y ∈ X : y ∼ x} is the equivalence class of x; it is denoted as [x]. Definition 20 Given an equivalence relation ∼ ...
Aalborg Universitet A convenient category for directed homotopy Fajstrup, Lisbeth; Rosický, J.
... X is I-generated if it has the property that a subset S ⊆ X is open if and only if f −1 (S) is open for every continuous map f : Z → X with Z ∈ I. Thus we get I-generated spaces of [7] in this case. We follow the terminology of [7] although it is somewhat misleading because, in the classical case of ...
... X is I-generated if it has the property that a subset S ⊆ X is open if and only if f −1 (S) is open for every continuous map f : Z → X with Z ∈ I. Thus we get I-generated spaces of [7] in this case. We follow the terminology of [7] although it is somewhat misleading because, in the classical case of ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.