On sigma-Induced L-Fuzzy Topological Spaces
... Conversely, let x,y∈X(x y) and p,q∈pr(L). From the fuzzy complete Hausdrofness of (X, ( T )), there exists basic open L-fuzzy sets , which are defined by respectively (z) = if z∈U, (z)=0 otherwise and (z)= if z V, (z)=0 otherwise, where U and V are regular -subsets in ( X,T ) and ∈L such that xp∈ , ...
... Conversely, let x,y∈X(x y) and p,q∈pr(L). From the fuzzy complete Hausdrofness of (X, ( T )), there exists basic open L-fuzzy sets , which are defined by respectively (z) = if z∈U, (z)=0 otherwise and (z)= if z V, (z)=0 otherwise, where U and V are regular -subsets in ( X,T ) and ∈L such that xp∈ , ...
Compact groups and products of the unit interval
... Remark 3 3 . We have shown in the proof of Theorem 3 1 that if X is a compact Hausdorff space, then (i) implies (ii), where (i) X has B™ as a subspace; (ii) there is a continuous mapping of X onto l m . Juhasz, ([8], theorem 3-18), shows that condition (ii) is equivalent to various other conditions. ...
... Remark 3 3 . We have shown in the proof of Theorem 3 1 that if X is a compact Hausdorff space, then (i) implies (ii), where (i) X has B™ as a subspace; (ii) there is a continuous mapping of X onto l m . Juhasz, ([8], theorem 3-18), shows that condition (ii) is equivalent to various other conditions. ...
this PDF file - European Journal of Pure and Applied
... then it is β-open in the usual sense. Indeed, if A is I − β-open, then there is a preopen set G such that G \ A,and A \ cl(G) ∈ I = {;}, and so G ⊆ A ⊆ cl(G), proving that A is β-open. Conversely, suppose that whenever a set A is I − β-open, then it is β-open. Let B ∈ I. Then, B is I − β-open, and b ...
... then it is β-open in the usual sense. Indeed, if A is I − β-open, then there is a preopen set G such that G \ A,and A \ cl(G) ∈ I = {;}, and so G ⊆ A ⊆ cl(G), proving that A is β-open. Conversely, suppose that whenever a set A is I − β-open, then it is β-open. Let B ∈ I. Then, B is I − β-open, and b ...
Local compactness - GMU Math 631 Spring 2011
... Examples 7. The following spaces are not locally compact: (1) Q;2 (2) RN ; (3) Cp (X) where X is any non-discrete Tychonoff space; (4) Niemytzky plane; (5) Zorgenfrey line; (6) The metric hedgehog; (7) The quotient hedgehog.3 ...
... Examples 7. The following spaces are not locally compact: (1) Q;2 (2) RN ; (3) Cp (X) where X is any non-discrete Tychonoff space; (4) Niemytzky plane; (5) Zorgenfrey line; (6) The metric hedgehog; (7) The quotient hedgehog.3 ...
