domains of perfect local homeomorphisms
... Question 1.2. Given a map p : X —►Y, what conditions on p, X , or Y (other than the obvious one) will assure that the domain X is a union of finitely many components? In particular, to what degree can the requirement of [3, 1.4], that p be proper, be relaxed? Specifically, if p : X —>Y is a perfect ...
... Question 1.2. Given a map p : X —►Y, what conditions on p, X , or Y (other than the obvious one) will assure that the domain X is a union of finitely many components? In particular, to what degree can the requirement of [3, 1.4], that p be proper, be relaxed? Specifically, if p : X —>Y is a perfect ...
BAIRE`S THEOREM AND ITS APPLICATIONS The completeness of
... T3 , ... are dense and open in X. Let W be any open set in X. We have to show that Vn has a point in W if W 6= ∅. Let ρ be a metric in X; let us write B(x, r) = {y ∈ X : ρ(x, y) < r} and let B̄(x, r) be the closure of B(x, r). Since V1 is dense, W ∩ V1 is a nonempty set, and we can therefore find x1 ...
... T3 , ... are dense and open in X. Let W be any open set in X. We have to show that Vn has a point in W if W 6= ∅. Let ρ be a metric in X; let us write B(x, r) = {y ∈ X : ρ(x, y) < r} and let B̄(x, r) be the closure of B(x, r). Since V1 is dense, W ∩ V1 is a nonempty set, and we can therefore find x1 ...
... Mashhour [2] introduced pre-open sets in a topological space and studied some of their properties. In 1979 kasahara [1] defined the operator α on atopological space (X,) as a map from P(X) to P(X) such that Uα (U) for every U . In 1991 Ogata[4] called the operation α as γ operation and introdu ...
Topological properties
... 2.1. Definition and first examples. Probably many of you have seen the notion of compact space in the context of subsets of Rn , as sets which are closed and bounded. Although not obviously at all, this is a topological property (it can be defined using open sets only). Definition 4.14. Given a topo ...
... 2.1. Definition and first examples. Probably many of you have seen the notion of compact space in the context of subsets of Rn , as sets which are closed and bounded. Although not obviously at all, this is a topological property (it can be defined using open sets only). Definition 4.14. Given a topo ...
e-7 Uniform Spaces, I - Analysis Group TU Delft
... {Bd (x, r ): x ∈ X } for some r > 0. In a topological group we get, as before, four types of uniform covers, each one defined by the requirement of having a refinement of the form L N = {x N : x ∈ G}, R N = {N x: x ∈ G}, T N = {x N ∩ N x: x ∈ G} or S N = {x N ∪ N x: x ∈ G}, respectively. Again, in a ...
... {Bd (x, r ): x ∈ X } for some r > 0. In a topological group we get, as before, four types of uniform covers, each one defined by the requirement of having a refinement of the form L N = {x N : x ∈ G}, R N = {N x: x ∈ G}, T N = {x N ∩ N x: x ∈ G} or S N = {x N ∪ N x: x ∈ G}, respectively. Again, in a ...
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.