(core) compactly generated spaces
... Let X and Y be topological spaces and let C(X, Y ) denote the set of continuous maps from X to Y . Given any continuous map f : A × X → Y , one has a function f : A → C(X, Y ) defined by f (a) = (x 7→ f (a, x)), called the exponential transpose of f . A topology on C(X, Y ) is said to be exponential ...
... Let X and Y be topological spaces and let C(X, Y ) denote the set of continuous maps from X to Y . Given any continuous map f : A × X → Y , one has a function f : A → C(X, Y ) defined by f (a) = (x 7→ f (a, x)), called the exponential transpose of f . A topology on C(X, Y ) is said to be exponential ...
tychonoff`s theorem - American Mathematical Society
... (B) A topological space X is compact if and only if for each collection of closed subsets of X with the finite intersection property (the intersection offinitely many elements of the collection is nonempty) the intersection of all elements of the collection is nonempty. Definition. Let E be a subset ...
... (B) A topological space X is compact if and only if for each collection of closed subsets of X with the finite intersection property (the intersection offinitely many elements of the collection is nonempty) the intersection of all elements of the collection is nonempty. Definition. Let E be a subset ...
ON WEAK FORMS OF PREOPEN AND PRECLOSED
... concept of weak preopenness as a natural dual to the weak precontinuity due to A. Kar and P. Bhattacharya [14]. Definition 2.1. A function f : (X, τ ) → (Y, σ) is said to be weakly preopen if f (U ) ⊂ pInt(f (Cl(U ))) for each open set U of X. Clearly, every weakly open function is weakly preopen an ...
... concept of weak preopenness as a natural dual to the weak precontinuity due to A. Kar and P. Bhattacharya [14]. Definition 2.1. A function f : (X, τ ) → (Y, σ) is said to be weakly preopen if f (U ) ⊂ pInt(f (Cl(U ))) for each open set U of X. Clearly, every weakly open function is weakly preopen an ...
Houston Journal of Mathematics
... If Si = U U(x) and Sz = (X \ clU) U{x}, then Si and & are semiopen, hene sg-open. By assumption, Si n S2 = {x} is sg-open, i.e. D = X \ {x} is sg-closed. ...
... If Si = U U(x) and Sz = (X \ clU) U{x}, then Si and & are semiopen, hene sg-open. By assumption, Si n S2 = {x} is sg-open, i.e. D = X \ {x} is sg-closed. ...
