Metrizable spaces are studied in which every closed set is an α-limit for some continuous map and point. It shown that this property enjoyed by space containing sufficiently many arcs (formalized the notion of a with enough arcs), though such need not be arcwise connected. Further it preserved topological sums, products images quotients. However, positive results do hold metrizable obtained tho...

The purpose of the present paper is to introduce the concept of generalized closed sets in Čech closure spaces and investigate some of their characterizations. 2000 Mathematics Subject Classification: 54A05.

in this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. by using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. moreover, we investigate the ...

In this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. By using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. Also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. Moreover, we investigate the ...

We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets.

The notion of immune sets is extended to closed sets and Π 1 classes in particular. We construct aΠ 1 class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A,B) of separating sets for A and B is immune. We show that every perfect thin Π 1 class is immune. We define the stronger notion of prompt immunity and construct an exampl...

We investigate notions of randomness in the space C[2] of nonempty closed subsets of {0, 1}. A probability measure is given and a version of the Martin-Löf Test for randomness is defined. Π 2 random closed sets exist but there are no random Π 1 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of 2 may be defined as the ...

A set is called r-closed left-r.e. iff every set r-reducible to it is also a left-r.e. set. It is shown that some but not all left-r.e. cohesive sets are many-one closed left-r.e. sets. Ascending reductions are many-one reductions via an ascending function; left-r.e. cohesive sets are also ascending closed left-r.e. sets. Furthermore, it is shown that there is a weakly 1-generic many-one closed...

‎Let $X$ be a real normed  space, then  $C(subseteq X)$  is  functionally  convex  (briefly, $F$-convex), if  $T(C)subseteq Bbb R $ is  convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)subseteq Bbb R $ is  closed  for all bounded linear transformations $Tin B(X,R)$. We improve the    Krein-Milman theorem  ...