$\ alpha r $-spaces and some of their properties
... Theorem 1. Let (X, p) be an r-space and let the relation of closure g of this r-space satisfy the condition (2a). Then (X, g) is a topological space. Proof. To prove that an r-space (X, p) is a topological space it suffices to-show that p is a closure operator on X (see [1]). Thus we must show that ...
... Theorem 1. Let (X, p) be an r-space and let the relation of closure g of this r-space satisfy the condition (2a). Then (X, g) is a topological space. Proof. To prove that an r-space (X, p) is a topological space it suffices to-show that p is a closure operator on X (see [1]). Thus we must show that ...
ON c-PRECONTINUOUS FUNCTIONS
... Theorem 3.6 : If f : X Y is c-precontinuous function and A be an α-open subset of X , then f/A : A Y is also c –precontinuous . Easy proof of the Theorem follows by Lemma – 3.5 above. Lemma 3.7 [ 17] : If V PO(X) and U SO(X) , then U V PO(U). Now, we give the following. Theorem 3.8 : If ...
... Theorem 3.6 : If f : X Y is c-precontinuous function and A be an α-open subset of X , then f/A : A Y is also c –precontinuous . Easy proof of the Theorem follows by Lemma – 3.5 above. Lemma 3.7 [ 17] : If V PO(X) and U SO(X) , then U V PO(U). Now, we give the following. Theorem 3.8 : If ...
spaces every quotient of which is metrizable
... Metrizability of quotients of metric spaces has been studied by many mathematicians [1, 2, 6, 7]. In an elementary course in general topology we learn that every continuous image in Hausdorff space of a compact metric space is metrizable [8]. In [7], Willard proved that every closed continuous image ...
... Metrizability of quotients of metric spaces has been studied by many mathematicians [1, 2, 6, 7]. In an elementary course in general topology we learn that every continuous image in Hausdorff space of a compact metric space is metrizable [8]. In [7], Willard proved that every closed continuous image ...
On Q*O compact spaces - Scitech Research Organisation
... [ 19 ] that if a connected compact Hausdorff space has a countable cover of pairwise disjoint closed sets , at most one of those sets is nonvoid In 1992, Cater and Daily showed that if a complete , connected , locally connected metric space is covered by countably many proper closed sets, then some ...
... [ 19 ] that if a connected compact Hausdorff space has a countable cover of pairwise disjoint closed sets , at most one of those sets is nonvoid In 1992, Cater and Daily showed that if a complete , connected , locally connected metric space is covered by countably many proper closed sets, then some ...
General Topology II - National Open University of Nigeria
... is a basis for the subspace topology in Y. Proof. Let U be an open set of X and y U Y, By definition of basis, there exists B B such that y B U. Then y B Y U Y. It follows from proposition 3.2 that BY is a basis for the subspace topology on Y. When dealing with a space X and a subspace Y of X, you n ...
... is a basis for the subspace topology in Y. Proof. Let U be an open set of X and y U Y, By definition of basis, there exists B B such that y B U. Then y B Y U Y. It follows from proposition 3.2 that BY is a basis for the subspace topology on Y. When dealing with a space X and a subspace Y of X, you n ...