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L-fuzzy compactness and related concepts
The compactness defined by Warner and McLean is extended to arbitrary L-fuzzy sets where L is a fuzzy lattice, i.e., a completely distributive lattice with an order reversing involution. It is shown that with our compactness we can build up a satisfactory theory. The different definitions of compactness in L-fuzzy topological spaces are stated and other characterizations of some of these notions are obtained. We also study their goodness and establish the inter-relations between the compactnesses which are good extensions.
Good definitions of L-fuzzy regularity and normality are proposed.
Following the lines of our compactness we suggest two definitions of L-fuzzy local compactness that are good extensions of the respective ordinary versions. A comparison between them is presented and some of their properties studied. A one point compactification is also obtained.
By introducing a new definition of a locally finite family of L-fuzzy sets and combining it with our definition of compactness, we propose an L-fuzzy paracompactness and study some of its properties.
Good definitions of L-fuzzy countable and sequential compactness and the Lindelof property are introduced and studied.
We also present, in L-fuzzy topological spaces, good extensions of S-closedness and RS-compactness. Some of their properties are examined.
Good L-fuzzy versions of almost compactness, near compactness and a strong compactness are put forward and studied. A comparison between these compactness related concepts is also presented
Sequential convergence in topological spaces
In this survey, my aim has been to discuss the use of sequences and countable
sets in general topology. In this way I have been led to consider five
different classes of topological spaces: first countable spaces, sequential
spaces, Frechet spaces, spaces of countable tightness and perfect spaces. We
are going to look at how these classes are related, and how well the various
properties behave under certain operations, such as taking subspaces, products,
and images under proper mappings. Where they are not well behaved we take the
opportunity to consider some relevant examples, which are often of special
interest. For instance, we examine an example of a Frechet space with unique
sequential limits that is not Hausdorff. I asked the question of whether there
exists in ZFC an example of a perfectly normal space that does not have
countable tightness: such an example was supplied and appears below. In our
discussion we shall report two independence theorems, one of which forms the
solution to the Moore-Mrowka problem. The results that we prove below include
characterisation theorems of sequential spaces and Frechet spaces in terms of
appropriate classes of continuous mappings, and the theorem that every
perfectly regular countably compact space has countable tightness.Comment: 29 pages. This version incorporates the correction of Proposition 3.2
to include an additional assumption (Hausdorff), whose necessity has been
pointed out by Alexander Gouberma
Connectedness modulo a topological property
Let be a topological property. We say that a space is
-connected if there exists no pair and of disjoint
cozero-sets of with non- closure such that the remainder
is contained in a cozero-set of with
closure. If is taken to be "being empty" then -connectedness coincides with connectedness in its usual sense. We
characterize completely regular -connected spaces, with
subject to some mild requirements. Then, we study conditions
under which unions of -connected subspaces of a space are
-connected. Also, we study classes of mappings which preserve
-connectedness. We conclude with a detailed study of the special
case in which is pseudocompactness. In particular, when
is pseudocompactness, we prove that a completely regular space
is -connected if and only if is connected, and that -connectedness is
preserved under perfect open continuous surjections. We leave some problems
open.Comment: 12 page
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