Semicontinuous functions and convexity
... Proof. Suppose that f is lower semicontinuous and xα → x. Say t < f (x). Because f is lower semicontinuous, f −1 (t, ∞] ∈ τ . As x ∈ f −1 (t, ∞] and xα → x, there is some αt such that α ≥ αt implies xα ∈ f −1 (t, ∞]. That is, if α ≥ αt then f (xα ) > t. This implies that lim inf f (xα ) ≥ t. But th ...
... Proof. Suppose that f is lower semicontinuous and xα → x. Say t < f (x). Because f is lower semicontinuous, f −1 (t, ∞] ∈ τ . As x ∈ f −1 (t, ∞] and xα → x, there is some αt such that α ≥ αt implies xα ∈ f −1 (t, ∞]. That is, if α ≥ αt then f (xα ) > t. This implies that lim inf f (xα ) ≥ t. But th ...
(A) Fuzzy Topological Spaces
... defined by γ = {0, 1, id[0,1] }. Let k be a real number, 0 ≤ k ≤ 1. The constant function f : X → [0, 1] with rule f (x) = k for x ∈ X is fuzzy continuous, and so f −1 (id[0,1] ) ∈ δ. But for x ∈ X, f −1 (id[0,1] )(x) = id[0,1] (f (x)) = id[0,1] (k) = k, whence the constant fuzzy set k in X belongs ...
... defined by γ = {0, 1, id[0,1] }. Let k be a real number, 0 ≤ k ≤ 1. The constant function f : X → [0, 1] with rule f (x) = k for x ∈ X is fuzzy continuous, and so f −1 (id[0,1] ) ∈ δ. But for x ∈ X, f −1 (id[0,1] )(x) = id[0,1] (f (x)) = id[0,1] (k) = k, whence the constant fuzzy set k in X belongs ...
A Crash Course in Topological Groups
... Any (arbitrary) direct product of these with the product topology. Note that, for example, (Z/2Z)ω is not discrete with the product topology (it is homeomorphic to the Cantor set). ...
... Any (arbitrary) direct product of these with the product topology. Note that, for example, (Z/2Z)ω is not discrete with the product topology (it is homeomorphic to the Cantor set). ...
on generalized closed sets
... It is obvious that in any topological space X, every sg-closed subset of X is gs-closed. In [20], the class of Tgs -spaces was introduced where a space X is called Tgs if every gs-closed subset of X is sg-closed. The following result exhibits the relationship between Tgs -spaces and T1/2 -spaces. Th ...
... It is obvious that in any topological space X, every sg-closed subset of X is gs-closed. In [20], the class of Tgs -spaces was introduced where a space X is called Tgs if every gs-closed subset of X is sg-closed. The following result exhibits the relationship between Tgs -spaces and T1/2 -spaces. Th ...
7 Complete metric spaces and function spaces
... In fact, we saw that if X is totally bounded, then any sequence in X has a Cauchy subsequence. (The converse is also true.) Corollary 7.21. Let (X, d) be complete. Then A ⊂ X is compact if, and only if, it is closed and totally bounded. A subset A ⊂ X, X a topological space, is said to be relatively ...
... In fact, we saw that if X is totally bounded, then any sequence in X has a Cauchy subsequence. (The converse is also true.) Corollary 7.21. Let (X, d) be complete. Then A ⊂ X is compact if, and only if, it is closed and totally bounded. A subset A ⊂ X, X a topological space, is said to be relatively ...
ABSOLUTELY CLOSED SPACES
... Let X^={% : % is an open nonconvergent ultrafilter on X). Let Jc= Xu X^ be the disjoint union, under the following topology: The basic open sets of 2 are those of the form G = G u GT where GT = iaU : <%e X", G e
... Let X^={% : % is an open nonconvergent ultrafilter on X). Let Jc= Xu X^ be the disjoint union, under the following topology: The basic open sets of 2 are those of the form G = G u GT where GT = iaU : <%e X", G e
COMPACTLY GENERATED SPACES Contents 1
... π : X × L → X takes closed sets to closed sets, and takes k-closed sets to k-closed sets. Proof. The first statement says that if C is closed in X × L, then π(C) = { x ∈ X | (x × L) ∩ C 6= ∅ } is closed in X, or equivalently that { x ∈ X | (x × L) ∩ C = ∅ } is open in X. This is equivalent to the “t ...
... π : X × L → X takes closed sets to closed sets, and takes k-closed sets to k-closed sets. Proof. The first statement says that if C is closed in X × L, then π(C) = { x ∈ X | (x × L) ∩ C 6= ∅ } is closed in X, or equivalently that { x ∈ X | (x × L) ∩ C = ∅ } is open in X. This is equivalent to the “t ...