... The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh[11]. Subsequently, Chang [6] defined the notion of fuzzy topology. Since then various aspects of general topology were investigated and carried out in fuzzy sense by several authors of this field. The local properties ...

QUOTIENT SPACE OF LMC

... functions on S, or as a collection of ultrafilters on S. What is certain and indisputable is the fact that filters play an important role in the study of Stone-Čech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactif ...

... functions on S, or as a collection of ultrafilters on S. What is certain and indisputable is the fact that filters play an important role in the study of Stone-Čech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactif ...

On Chains in H-Closed Topological Pospaces

... minimal element y ∈ X and a maximal element z ∈ X such that y x z (cf. [10]). Every maximal chain in a compact topological pospace is a compact subset and hence it contains minimal and maximal elements. Also, for any point x of a compact topological pospace X there exists a base at x which consi ...

... minimal element y ∈ X and a maximal element z ∈ X such that y x z (cf. [10]). Every maximal chain in a compact topological pospace is a compact subset and hence it contains minimal and maximal elements. Also, for any point x of a compact topological pospace X there exists a base at x which consi ...

MAXIMAL QHC-SPACES 1. Introduction and Background. In 1924

... For a topological space (X, r) the relative topology on a subset A of X is T | A, the collection of open neighborhoods for a point x is Q/T(x), and the product of (Xa, r j for a G

... For a topological space (X, r) the relative topology on a subset A of X is T | A, the collection of open neighborhoods for a point x is Q/T(x), and the product of (Xa, r j for a G

An introduction to classical descriptive set theory

... 3. Borel sets A σ-algebra is a family of sets which is closed under complements, countable intersections, and countable unions. A subset of a topological space is Borel if it is in the smallest σ-algebra containing the open sets. While the structure of Borel sets is a central focus of classical desc ...

... 3. Borel sets A σ-algebra is a family of sets which is closed under complements, countable intersections, and countable unions. A subset of a topological space is Borel if it is in the smallest σ-algebra containing the open sets. While the structure of Borel sets is a central focus of classical desc ...

Modal compact Hausdorff spaces

... they need not preserve order. Also, isomorphisms need not be homomorphisms with respect to the modal operators involved, and it is possible to have two different modal de Vries operators on the same de Vries algebra giving isomorphic modal de Vries algebras. We identify two full subcategories of MDV ...

... they need not preserve order. Also, isomorphisms need not be homomorphisms with respect to the modal operators involved, and it is possible to have two different modal de Vries operators on the same de Vries algebra giving isomorphic modal de Vries algebras. We identify two full subcategories of MDV ...

Topology I with a categorical perspective

... modern, categorical point of view. There are a number of reasons this alternative can be better. Since many students are familiar with point-set ideas already, they are in a good position to learn something new about these ideas, like the universal properties characterizing them. Plus, using categor ...

... modern, categorical point of view. There are a number of reasons this alternative can be better. Since many students are familiar with point-set ideas already, they are in a good position to learn something new about these ideas, like the universal properties characterizing them. Plus, using categor ...

Lecture Notes (unique pdf file)

... and the topology τi having {(−∞, a) : a ∈ R} as a basis are incomparable. In fact, it is easy to see that τi = {(−∞, a) : a ∈ R} ∪ {∅, R} as these are the unions of sets in the given basis. In particular, we have that R − {0} is in τc but not τi . Moreover, we have that (−∞, 0) is in τi but not τc . ...

... and the topology τi having {(−∞, a) : a ∈ R} as a basis are incomparable. In fact, it is easy to see that τi = {(−∞, a) : a ∈ R} ∪ {∅, R} as these are the unions of sets in the given basis. In particular, we have that R − {0} is in τc but not τi . Moreover, we have that (−∞, 0) is in τi but not τc . ...

A survey of ultraproduct constructions in general topology

... Theorem 2.4 (Keisler’s Ultrapower Theorem: [28, 77]). Let λ be an infinite cardinal where the GCH holds (i.e., 2λ = λ+ ), and let I be a set whose cardinality is λ. Then there is an ultrafilter D on I such that if L is an alphabet with at most λ symbols, and if A and B are elementarily equivalent L- ...

... Theorem 2.4 (Keisler’s Ultrapower Theorem: [28, 77]). Let λ be an infinite cardinal where the GCH holds (i.e., 2λ = λ+ ), and let I be a set whose cardinality is λ. Then there is an ultrafilter D on I such that if L is an alphabet with at most λ symbols, and if A and B are elementarily equivalent L- ...

NON-HAUSDORFF GROUPOIDS, PROPER ACTIONS AND K

... an isomorphism for amenable groupoids. Certainly the first step is to define the notion of proper groupoid (since an action of a groupoid G on a space Z is proper if and only if the crossedproduct groupoid Z o G is proper). Our definition is as follows: a topological groupoid G is proper if the map ...

... an isomorphism for amenable groupoids. Certainly the first step is to define the notion of proper groupoid (since an action of a groupoid G on a space Z is proper if and only if the crossedproduct groupoid Z o G is proper). Our definition is as follows: a topological groupoid G is proper if the map ...

1. The Baire category theorem

... A set A ⊆ S is of second category when it is not of the first category. Via the Baire category theorem we will become able to give many examples - for example show that R is of the second category as a subset of itself. (As well as [0, 1] ⊆ R). Subsequently we shall then see why the notion is useful ...

... A set A ⊆ S is of second category when it is not of the first category. Via the Baire category theorem we will become able to give many examples - for example show that R is of the second category as a subset of itself. (As well as [0, 1] ⊆ R). Subsequently we shall then see why the notion is useful ...

Topological properties

... Proof. To prove the proposition we will use the following Exercise 4.2. If A, B ⊂ X are connected and A ∩ B 6= ∅, then A ∪ B is connected. For (i) of the proposition, let x ∈ X and define C(x) as the union of all connected subsets of X containing x. We claim that C(x) is a connected component. The o ...

... Proof. To prove the proposition we will use the following Exercise 4.2. If A, B ⊂ X are connected and A ∩ B 6= ∅, then A ∪ B is connected. For (i) of the proposition, let x ∈ X and define C(x) as the union of all connected subsets of X containing x. We claim that C(x) is a connected component. The o ...

General Topology - Faculty of Physics University of Warsaw

... 6.5 Basic constructions in metric spaces . ...

... 6.5 Basic constructions in metric spaces . ...

CLASSIFYING THE TYPES OF PRINCIPAL GROUPOID C

... We can weaken the conditions in Proposition 3.8 and show that, for principal groupoids, ψ is a homeomorphism when G0 /G is a T0 space. In doing this, we actually describe the ideal structure of the associated groupoid C ∗ -algebra. We will also prove a generalization of Gootman’s Theorem 1.1 for pri ...

... We can weaken the conditions in Proposition 3.8 and show that, for principal groupoids, ψ is a homeomorphism when G0 /G is a T0 space. In doing this, we actually describe the ideal structure of the associated groupoid C ∗ -algebra. We will also prove a generalization of Gootman’s Theorem 1.1 for pri ...

Spaces in which compact subsets are closed and the lattice of $ T_1

... is then a partial positive answer to the above-mentioned question of Larson. Theorem 10. Every minimal KC-topology on a countable set is compact. These results should be contrasted with the case of minimal Hausdorff spaces. An example of a countable minimal Hausdorff space which is not countably com ...

... is then a partial positive answer to the above-mentioned question of Larson. Theorem 10. Every minimal KC-topology on a countable set is compact. These results should be contrasted with the case of minimal Hausdorff spaces. An example of a countable minimal Hausdorff space which is not countably com ...

Topologies making a given ideal nowhere dense or meager

... Let X be a set and let Y be an ideal on X. In this paper we show how to find a topology T on X such that r-nowhere dense (or r-meager) sets are exactly the sets in 2. We try to find the “best” possible topology with such property. In Section 1 we discuss the ideals ($1) and 9(X). We also show that f ...

... Let X be a set and let Y be an ideal on X. In this paper we show how to find a topology T on X such that r-nowhere dense (or r-meager) sets are exactly the sets in 2. We try to find the “best” possible topology with such property. In Section 1 we discuss the ideals ($1) and 9(X). We also show that f ...

Characterizing continuous functions on compact

... discusses the notion of the resolution of a space by a family of spaces, resolutions being our main tool for combining the various constructions in the sequel. Although, in general, resolutions do not combine well with continuity, Theorem 5.4 provides us with natural conditions allowing us to glue o ...

... discusses the notion of the resolution of a space by a family of spaces, resolutions being our main tool for combining the various constructions in the sequel. Although, in general, resolutions do not combine well with continuity, Theorem 5.4 provides us with natural conditions allowing us to glue o ...

Some results on sequentially compact extensions

... consisting of Tychonoff sequential compactifications of X. (As we have already remarked, those sets can be empty.) In the first section of this paper we observe that, for Tychonoff LSC spaces, oX can fail to be Tychonoff. However, we prove that every Tychonoff LSC space has a Tychonoff one-point seq ...

... consisting of Tychonoff sequential compactifications of X. (As we have already remarked, those sets can be empty.) In the first section of this paper we observe that, for Tychonoff LSC spaces, oX can fail to be Tychonoff. However, we prove that every Tychonoff LSC space has a Tychonoff one-point seq ...

G. Bezhanishvili. Zero-dimensional proximities and zero

... Axiom (P4) is the most fundamental among the above six axioms. Because of its importance, some authors call it the strong axiom (see, e.g., [13, p. 9]). In the next section we will strengthen it even further in order to axiomatize zerodimensional proximities. Note that some authors (including Naimpa ...

... Axiom (P4) is the most fundamental among the above six axioms. Because of its importance, some authors call it the strong axiom (see, e.g., [13, p. 9]). In the next section we will strengthen it even further in order to axiomatize zerodimensional proximities. Note that some authors (including Naimpa ...

covariant and contravariant approaches to topology

... T1. Then, X is T2 (Hausdorff) iff S is an absolute neighborhood extensor of X with respect to finite subspaces. 4. Suppose X is To. Then, X is Ta1/2 (Tychonoff) iff the topology of X is the contravariant topology induced by a family of functions { f X I} s. 5. Suppose X is TI. Then, X is T4 (normal) ...

... T1. Then, X is T2 (Hausdorff) iff S is an absolute neighborhood extensor of X with respect to finite subspaces. 4. Suppose X is To. Then, X is Ta1/2 (Tychonoff) iff the topology of X is the contravariant topology induced by a family of functions { f X I} s. 5. Suppose X is TI. Then, X is T4 (normal) ...

LECTURE NOTES IN TOPOLOGICAL GROUPS 1

... (1) the inversion map, left and right translations, all conjugations, are homeomorphisms G → G. (2) G is homogeneous as a topological space. (3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that f (x) = y and f (y) = x. (4) Which of the following topological spaces are of the gr ...

... (1) the inversion map, left and right translations, all conjugations, are homeomorphisms G → G. (2) G is homogeneous as a topological space. (3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that f (x) = y and f (y) = x. (4) Which of the following topological spaces are of the gr ...

PROPERTIES OF H-SETS, KAT ˇETOV SPACES AND H

... closed spaces and H-sets as well – with some subtle but important differences. As a preview, consider these properties parallel to the properties of compact spaces above. • A regular closed subset of an H-set is H-closed. • A θ -continuous image of an H-set is an H-set. • A product of H-closed spac ...

... closed spaces and H-sets as well – with some subtle but important differences. As a preview, consider these properties parallel to the properties of compact spaces above. • A regular closed subset of an H-set is H-closed. • A θ -continuous image of an H-set is an H-set. • A product of H-closed spac ...

BORNOLOGICAL CONVERGENCES A. Lechicki, S. Levi, and A

... where B(A, ε) is the ε-enlargement of the set A of radius ε. The Hausdorff disH tance induces a convergence H on the power set 2X by defining At → A whenever h(At , A) → 0. However, this convergence works well only when restricted to bounded (closed) subsets. For unbounded sets convergence in the Ha ...

... where B(A, ε) is the ε-enlargement of the set A of radius ε. The Hausdorff disH tance induces a convergence H on the power set 2X by defining At → A whenever h(At , A) → 0. However, this convergence works well only when restricted to bounded (closed) subsets. For unbounded sets convergence in the Ha ...

# Felix Hausdorff

Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.