SOFT TOPOLOGICAL QUESTIONS AND ANSWERS M. Matejdes
... Remark 3.1. It is clear, intθ (A) ⊂ int(A) and intθ (A) is open. Consequently, τθ ⊂ τ . Similarly, cl(A) ⊂ clθ (S) and clθ (A) is closed. Proposition 3.2. ([3]) A family τθ of all θ-open sets in a topological space (X, τ ) forms a topology. The next proposition characterizes the equality τ = τθ . Pr ...
... Remark 3.1. It is clear, intθ (A) ⊂ int(A) and intθ (A) is open. Consequently, τθ ⊂ τ . Similarly, cl(A) ⊂ clθ (S) and clθ (A) is closed. Proposition 3.2. ([3]) A family τθ of all θ-open sets in a topological space (X, τ ) forms a topology. The next proposition characterizes the equality τ = τθ . Pr ...
(A) Fuzzy Topological Spaces
... First Obstacle: A product of arbitrarily many compace fuzzy topological spaces need not be compact. Another Obstacle: Some constant functions from one fuzzy topological space to another fail to be continuous. Proposition. Let (X, δ) be a fuzzy topological space. Then every constant function from (X ...
... First Obstacle: A product of arbitrarily many compace fuzzy topological spaces need not be compact. Another Obstacle: Some constant functions from one fuzzy topological space to another fail to be continuous. Proposition. Let (X, δ) be a fuzzy topological space. Then every constant function from (X ...
Strongly g -Closed Sets in Topological Spaces 1 Introduction
... Theorem 3.5. A set A is strongly g ∗ -closed iff cl(int(A))−A contains no non empty closed set. Proof. Necessary : Suppose that F is non empty closed subset of cl(int(A)). Now F ⊆ cl(int(A)) − A implies F ⊆ cl(int(A)) ∩ Ac , since cl(int(A)) − A = cl(int(A)) ∩ Ac . Thus F ⊆ cl(int(A)). Now F ⊆ Ac im ...
... Theorem 3.5. A set A is strongly g ∗ -closed iff cl(int(A))−A contains no non empty closed set. Proof. Necessary : Suppose that F is non empty closed subset of cl(int(A)). Now F ⊆ cl(int(A)) − A implies F ⊆ cl(int(A)) ∩ Ac , since cl(int(A)) − A = cl(int(A)) ∩ Ac . Thus F ⊆ cl(int(A)). Now F ⊆ Ac im ...
Topology of the Real Numbers
... Example 5.35. The set N is closed, but it is not compact. The sequence (n) in N has no convergent subsequence since every subsequence diverges to infinity. As these examples illustrate, a compact set must be closed and bounded. Conversely, the Bolzano-Weierstrass theorem implies that that every clos ...
... Example 5.35. The set N is closed, but it is not compact. The sequence (n) in N has no convergent subsequence since every subsequence diverges to infinity. As these examples illustrate, a compact set must be closed and bounded. Conversely, the Bolzano-Weierstrass theorem implies that that every clos ...
![arXiv:math/0412558v2 [math.GN] 10 Apr 2016](http://s1.studyres.com/store/data/000780140_1-7e82978ab888d179066aebd70a132571-300x300.png)
Pages 31-40 - The Graduate Center, CUNY
... (3.7.1) We will show that when one takes K to be the category of sets, the inverse image by ψ of every K-valued presheaf G must exist (the notation and hypotheses for X, Y , ψ being those of (3.5.3)). Indeed, for every open U ⊆ X, G 0 (U ) is defined as follows: an element s0 of G 0 (U ) is a family ...
... (3.7.1) We will show that when one takes K to be the category of sets, the inverse image by ψ of every K-valued presheaf G must exist (the notation and hypotheses for X, Y , ψ being those of (3.5.3)). Indeed, for every open U ⊆ X, G 0 (U ) is defined as follows: an element s0 of G 0 (U ) is a family ...
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.