A base B of a space X is called an OIF base when every element of B is a subset of only a finite number of other elements of B. We will explore the fundamental properties of spaces having such bases. In particular, we will show that in T2 spaces, strong OIF bases are the same as uniform bases, and that in T3 spaces where all subspaces have OIF bases, compactness, countable compactness, or local...

Motivated by research on hyperfinite equivalence relations we define a coloring property for countable groups. We prove that every countable group has the coloring property. This implies a compactness theorem for closed complete sections of the free part of the shift action of G on 2G . Our theorems generalize known results about Z.

The concepts of γ-compactness, countable γ-compactness, the γ-Lindelöf property are introduced in L-topological spaces by means of γ-open L-sets and their inequalities when L is a complete DeMorgan algebra. These definitions do not rely on the structure of the basis lattice L and no distributivity in L is required.

We show that a graph can always be decomposed into edge-disjoint subgraphs of countable cardinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinals. We also show that the vertex set of any graph can be endowed with a well-ordering which has a certain compactness property with respect to edge-separation.

A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...

Abstract In this note we show that proof-theoretic uniform boundedness or bounded collection principles which allow one to formalize certain instances of countable Heine–Borel compactness in proofs using abstract metric structures must be carefully distinguished from an unrestricted use compactness.

In this paper, a new notion of sequential compactness is introduced in L-topological spaces, which is called sequentially S∗-compactness. If L = [0, 1], sequential ultra-compactness, sequential N-compactness and sequential strong compactness imply sequential S∗-compactness, and sequential S∗-compactness implies sequential F-compactness. The intersection of a sequentially S∗-compact L-set and a ...

Abstract In this paper, we introducing fuzzy neutrosophic supra first countable (FNS - FCS), second (FNS- SCS) and compactness compactness), Q μ compactness) in topological space (FNSTS). derive the union of two FNS – compact spaces is also similarly space. Also define some theorems using finite intersection property productive property. Finally observe that our notions preserve under one one, ...

In this paper several results of N. Noble concerning closed projections are extended using generalizations of first countable, Frechet, and sequential spaces. We also consider compactness conditions defined by requiring that certain nets have cluster points.