Ideal Resolvability - Mathematics TU Graz
... A topological space (X, τ, I) is called maximal I-resolvable if (X, τ, I) is I-resolvable and (X, σ, I) is not I-resolvable for every topology σ which is strictly finer than τ . Note that {∅}-resolvable spaces are called maximal resolvable. However, often in the literature the term maximally resolva ...
... A topological space (X, τ, I) is called maximal I-resolvable if (X, τ, I) is I-resolvable and (X, σ, I) is not I-resolvable for every topology σ which is strictly finer than τ . Note that {∅}-resolvable spaces are called maximal resolvable. However, often in the literature the term maximally resolva ...
Extension and Selection theorems in Topological spaces
... of the family. A topological space is supercompact if it has a binary subbase 13: any non empty subfamily F C B such that any two members of F meet has a non empty intersection. A family B is normal if for any pair Bi , B2 E ~3 such that Bi D B2 = 0 another pair Bi, B2 E B can be found such that Bi ...
... of the family. A topological space is supercompact if it has a binary subbase 13: any non empty subfamily F C B such that any two members of F meet has a non empty intersection. A family B is normal if for any pair Bi , B2 E ~3 such that Bi D B2 = 0 another pair Bi, B2 E B can be found such that Bi ...
GO-SPACES WITH σ-CLOSED DISCRETE DENSE SUBSETS 1
... For each n and each G ∈ Gn , let G∗ = {S(p) : p ∈ X and S(p) ⊆ G}. Let Ln = {f [G∗ ] : G ∈ Gn }. Because f is a closed mapping, each member of Ln is open in Y . It is easy to check that hLn i is a Gδ -diagonal sequence of open covers of Y . Because f is continuous and closed, Y is paracompact and pe ...
... For each n and each G ∈ Gn , let G∗ = {S(p) : p ∈ X and S(p) ⊆ G}. Let Ln = {f [G∗ ] : G ∈ Gn }. Because f is a closed mapping, each member of Ln is open in Y . It is easy to check that hLn i is a Gδ -diagonal sequence of open covers of Y . Because f is continuous and closed, Y is paracompact and pe ...
Products of completion regular measures
... 3. To help give meaning to the concept of “quasi-dyadic” space, we give some elementary properties of these spaces. Proposition. (a) A continuous image of a quasi-dyadic space is quasidyadic. Any product of quasi-dyadic spaces is quasi-dyadic. (b) A space with countable network is quasi-dyadic. (c) ...
... 3. To help give meaning to the concept of “quasi-dyadic” space, we give some elementary properties of these spaces. Proposition. (a) A continuous image of a quasi-dyadic space is quasidyadic. Any product of quasi-dyadic spaces is quasi-dyadic. (b) A space with countable network is quasi-dyadic. (c) ...
Topologies on function spaces and hyperspaces
... C(Y, Z) is admissible if and only if for every space X, the continuity of a map h : X → Ct (Y, Z) implies that of the map h⋄ : X × Y → Z If in the above definitions it is assumed that the space X belongs to a fixed class A of topological spaces, then the topology t is called A-splitting or A-admissi ...
... C(Y, Z) is admissible if and only if for every space X, the continuity of a map h : X → Ct (Y, Z) implies that of the map h⋄ : X × Y → Z If in the above definitions it is assumed that the space X belongs to a fixed class A of topological spaces, then the topology t is called A-splitting or A-admissi ...
FULL TEXT
... of sg-closed sets is sg-closed (we will show that this is not true even for two sets). We recall some definitions. A set A is called semi-open if A ⊆ cl(int(A)) and semiclosed if int(cl(A)) ⊆ A. The semi-interior (resp. semi-kernel ) of A, denoted by sint(A) (resp. sker(A), is the union (resp. inter ...
... of sg-closed sets is sg-closed (we will show that this is not true even for two sets). We recall some definitions. A set A is called semi-open if A ⊆ cl(int(A)) and semiclosed if int(cl(A)) ⊆ A. The semi-interior (resp. semi-kernel ) of A, denoted by sint(A) (resp. sker(A), is the union (resp. inter ...
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.