6.
... P1, P2, ….. Pn are semi open in X, Po is a *-semi open set with X \ Po an H-set. We may also choose each Pi Po, for i = 1, ….. n in such a basic semi open set. Definition 3.9 A topological space is a sTo space iff for each pair x and y of distinct points, there is a semineighbourhood of one point ...
... P1, P2, ….. Pn are semi open in X, Po is a *-semi open set with X \ Po an H-set. We may also choose each Pi Po, for i = 1, ….. n in such a basic semi open set. Definition 3.9 A topological space is a sTo space iff for each pair x and y of distinct points, there is a semineighbourhood of one point ...
Garrett 12-07-2011 1 Fujisaki’s Compactness Lemma and corollaries:
... Let Ξ = (C − C)2 ∩ k × be this finite set. Paraphrasing: given α ∈ 1 , there are a ∈ k × and ξ ∈ Ξ (ξ = ab above) such that (a · α−1 , (a · α−1 )−1 ) ∈ (C − C) × ξ −1 (C − C). ...
... Let Ξ = (C − C)2 ∩ k × be this finite set. Paraphrasing: given α ∈ 1 , there are a ∈ k × and ξ ∈ Ξ (ξ = ab above) such that (a · α−1 , (a · α−1 )−1 ) ∈ (C − C) × ξ −1 (C − C). ...
compactness on bitopological spaces
... compactness" and we study the properties of this spaces, also we define the continuous functions between these spaces. 1.Introduction The concept of "bitopological space" was introduced by Kelly [1] in 1963. A set equipped with two topologies is called a "bitopological space" and denote by (X, , whe ...
... compactness" and we study the properties of this spaces, also we define the continuous functions between these spaces. 1.Introduction The concept of "bitopological space" was introduced by Kelly [1] in 1963. A set equipped with two topologies is called a "bitopological space" and denote by (X, , whe ...
Course 212: Academic Year 1991-2 Section 4: Compact Topological
... of open sets in X covering A, there exists a finite collection V1 , V2 , . . . , Vr of open sets belonging to U such that A ⊂ V1 ∪ V2 ∪ · · · ∪ Vr . Proof A subset B of A is open in A (with respect to the subspace topology on A) if and only if B = A ∩ V for some open set V in X. The desired result t ...
... of open sets in X covering A, there exists a finite collection V1 , V2 , . . . , Vr of open sets belonging to U such that A ⊂ V1 ∪ V2 ∪ · · · ∪ Vr . Proof A subset B of A is open in A (with respect to the subspace topology on A) if and only if B = A ∩ V for some open set V in X. The desired result t ...
THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE 1
... In this paper we generalize Pospišil’s inequalities for the class of all topological spaces and Arhangel0 skiı̆’s inequality for the class of all T1 -topological spaces and show that Arhangel0 skiı̆’s inequality is true for a very large class of T1 -spaces. 2. The cardinal function nh(X) We begin w ...
... In this paper we generalize Pospišil’s inequalities for the class of all topological spaces and Arhangel0 skiı̆’s inequality for the class of all T1 -topological spaces and show that Arhangel0 skiı̆’s inequality is true for a very large class of T1 -spaces. 2. The cardinal function nh(X) We begin w ...
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.