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... hereditarily compact. What we want to show here is the following: If the product space of an arbitrary family of spaces is sg-compact, then all but one factor spaces must be finite and the remaining one must be (at most) sg-compact. Maki, Balachandran and Devi [14, Theorem 3,7] showed (under the add ...
... hereditarily compact. What we want to show here is the following: If the product space of an arbitrary family of spaces is sg-compact, then all but one factor spaces must be finite and the remaining one must be (at most) sg-compact. Maki, Balachandran and Devi [14, Theorem 3,7] showed (under the add ...
Universal real locally convex linear topological spaces
... 5° for every a; and U there exists X such that Hc^k.U. Those sets t7, V, ...will be termed v. N e u m a n n ' s n e i g h b o r h o o d s (N. nbhds). Two systems U', U" o f N . nbhds are said to be e q u i v a l e n t if for every UW there exists U'^Vi" with U" c U' and for every UW there exists U[^ ...
... 5° for every a; and U there exists X such that Hc^k.U. Those sets t7, V, ...will be termed v. N e u m a n n ' s n e i g h b o r h o o d s (N. nbhds). Two systems U', U" o f N . nbhds are said to be e q u i v a l e n t if for every UW there exists U'^Vi" with U" c U' and for every UW there exists U[^ ...
MORE ON ALMOST STRONGLY-θ-β-CONTINUOUS
... and Cl(A) respectively. A subset A of a space X is said to be regular open if A = Int(Cl(A)) and it’s complement is said to be regular closed if A = Cl(Int(A)). A subset A of a space X is called semi-open [10] (resp. β-open [1]) if A ⊆ Cl(Int(A)) (resp. A ⊆ Cl(Int(Cl(A)))). The family of all βopen s ...
... and Cl(A) respectively. A subset A of a space X is said to be regular open if A = Int(Cl(A)) and it’s complement is said to be regular closed if A = Cl(Int(A)). A subset A of a space X is called semi-open [10] (resp. β-open [1]) if A ⊆ Cl(Int(A)) (resp. A ⊆ Cl(Int(Cl(A)))). The family of all βopen s ...
On Almost Regular Spaces
... ⇒ V and X – V are disjoint . ⇒ X is almost regular . Theorem 3.3 : Let X and Y be nonempty topological spaces . The product X × Y is almost regular ⇔ both X and Y are almost regular . Proof : Step 1 : Suppose that X × Y is almost regular . Let x0 ∈ X . Let U be any open neighbourhood of x0 . Let y0 ...
... ⇒ V and X – V are disjoint . ⇒ X is almost regular . Theorem 3.3 : Let X and Y be nonempty topological spaces . The product X × Y is almost regular ⇔ both X and Y are almost regular . Proof : Step 1 : Suppose that X × Y is almost regular . Let x0 ∈ X . Let U be any open neighbourhood of x0 . Let y0 ...
The fundamental groupoid as a topological
... As a consequence of Theorem 5 (i) we have a transgression function A:n(X, x)-+G{x}. Corollary 6. Let X be path-connected and x e X. The following conditions are equivalent. (i) StGx is path-connected. (ii) A: n(X, x)-+G{x} is surjective. (iii) 5* : n(G, lx)^>n(X, x) is surjective. Proof. These follo ...
... As a consequence of Theorem 5 (i) we have a transgression function A:n(X, x)-+G{x}. Corollary 6. Let X be path-connected and x e X. The following conditions are equivalent. (i) StGx is path-connected. (ii) A: n(X, x)-+G{x} is surjective. (iii) 5* : n(G, lx)^>n(X, x) is surjective. Proof. These follo ...
LECtURE 7: SEPtEmBER 17 Closed sets and compact sets. Last
... We want to show that 1 ∈ S, because this will mean that [0, 1] can be covered by finitely many sets in U . Using the least upper bound property of R, define s = sup S. Now the argument proceeds in three steps: Step 1. Every point t with 0 ≤ t < s has to belong to S. Suppose that we had t �∈ S. Then ...
... We want to show that 1 ∈ S, because this will mean that [0, 1] can be covered by finitely many sets in U . Using the least upper bound property of R, define s = sup S. Now the argument proceeds in three steps: Step 1. Every point t with 0 ≤ t < s has to belong to S. Suppose that we had t �∈ S. Then ...
Reflexive cum coreflexive subcategories in topology
... The notions of reflexive and coreflexive subcategories in topology have received much attention in the recent past. (See e.g. Kennison [5], Herrlich [2], Herrlich and Strecker [4], Kannan [6-8].) In this paper we are concerned with the following question and its analogues: Let ~-be the category of a ...
... The notions of reflexive and coreflexive subcategories in topology have received much attention in the recent past. (See e.g. Kennison [5], Herrlich [2], Herrlich and Strecker [4], Kannan [6-8].) In this paper we are concerned with the following question and its analogues: Let ~-be the category of a ...
6. Compactness
... The description of topology as “rubber-sheet geometry” can be made precise by picturing map(X, Y ). We want to describe a deformation of one mapping into another. If f and g are in map(X, Y ), then a path in map(X, Y ) joining f and g is a mapping λ: [0, 1] → map(X, Y ) with λ(0) = f and λ(1) = g. T ...
... The description of topology as “rubber-sheet geometry” can be made precise by picturing map(X, Y ). We want to describe a deformation of one mapping into another. If f and g are in map(X, Y ), then a path in map(X, Y ) joining f and g is a mapping λ: [0, 1] → map(X, Y ) with λ(0) = f and λ(1) = g. T ...