of almost compact spaces - American Mathematical Society
... 1. Introduction. Amongst the most beautiful representation theorems for topological spaces are the characterization of compact Hausdorff spaces as closed subspaces of products of copies of the unit interval [0,1] (with its usual topology) and the characterization of realcompact spaces as closed subs ...
... 1. Introduction. Amongst the most beautiful representation theorems for topological spaces are the characterization of compact Hausdorff spaces as closed subspaces of products of copies of the unit interval [0,1] (with its usual topology) and the characterization of realcompact spaces as closed subs ...
OPEN, CLOSED, REGULARLY OPEN, REGULARLY CLOSED
... ( X , T ) has property P, where P can be each of the properties cited above [1]. The results above raised the question of whether or not topological properties could be further characterized using only certain types of sets within the space. With the important role of open and closed sets in the stu ...
... ( X , T ) has property P, where P can be each of the properties cited above [1]. The results above raised the question of whether or not topological properties could be further characterized using only certain types of sets within the space. With the important role of open and closed sets in the stu ...
Separation of Fuzzy Topological Space
... From the above theorems and definitions it is clear that A fuzzy set in a fuzzy topological space is said to be properly compact if every proper open cover of a fuzzy set is reducible to a finite proper open sub cover and every properly compact set in a fuzzy T2-space is closed. Thus the definition ...
... From the above theorems and definitions it is clear that A fuzzy set in a fuzzy topological space is said to be properly compact if every proper open cover of a fuzzy set is reducible to a finite proper open sub cover and every properly compact set in a fuzzy T2-space is closed. Thus the definition ...
Spectra of Small Categories and Infinite Loop Space Machines
... them. The notion of weak homotopy (studied in [13, 14]) is related to the nerve (or classifying space) of the categories: Two functors f and g are homotopic iff B f and B g are homotopic continuous maps. Here, B: C at → T op means the classical classifying space functor which is the composition of t ...
... them. The notion of weak homotopy (studied in [13, 14]) is related to the nerve (or classifying space) of the categories: Two functors f and g are homotopic iff B f and B g are homotopic continuous maps. Here, B: C at → T op means the classical classifying space functor which is the composition of t ...
ON P AND WEAKLY-P SPACES M. Khan, T. Noiri and B. Ahmad 1
... Throughout this paper, spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let X be a topological space and A be a subset of X . The closure of A and the interior of A in X are denoted by Cl(A) and Int(A), respectively. A subset A of X is said to ...
... Throughout this paper, spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let X be a topological space and A be a subset of X . The closure of A and the interior of A in X are denoted by Cl(A) and Int(A), respectively. A subset A of X is said to ...
The Baire Category Theorem
... 4. If G is open and dense in E, then {G is nowhere dense. 5. If F is closed and nowhere dense, then {F is dense. Exercise 3 Prove the five properties stated in Proposition 2. The Baire Category Theorem can be stated a second way as follows. Theorem 2 (Baire Category Theorem) A complete metric space ...
... 4. If G is open and dense in E, then {G is nowhere dense. 5. If F is closed and nowhere dense, then {F is dense. Exercise 3 Prove the five properties stated in Proposition 2. The Baire Category Theorem can be stated a second way as follows. Theorem 2 (Baire Category Theorem) A complete metric space ...
A Demonstration that Quotient Spaces of Locally Compact Hausdorff
... Definition 2.6. A collection A of subsets of a space X is said to cover X, or to be a covering of X, if the union of the elements of A is equal to X. It is called an open covering of X if its elements are open subsets of X. A subcovering is a subset C of A which is also a covering of X. Definition ...
... Definition 2.6. A collection A of subsets of a space X is said to cover X, or to be a covering of X, if the union of the elements of A is equal to X. It is called an open covering of X if its elements are open subsets of X. A subcovering is a subset C of A which is also a covering of X. Definition ...
