Some New Completeness Properties in Topological Spaces

One of the most widely known completeness property is the completeness of metric spaces and the other one being of a topological space in the sense of Cech. It is well known that a metrizable space X is completely metrizable if and only if X is Cech-complete. One of the generalisations of completeness of metric spaces is subcompactness. It has been established that, for metrizable spaces, subcompactness is equivalent to Cech-completeness. Also the concept of domain representability can be considered as a completeness property. In [1], Bennett and Lutzer proved that Cech-complete spaces are domain representable. They also proved, in [2], that subcompact regular spaces are domain representable. Then Fleissner and Yengulalp, in [3], gave a simplified characterization of domain representability. In this work, we introduce the completeness of a quasi-pair-base and study the topological spaces having such a base. Our results include the fact that Cech-complete spaces and subcompact spaces have complete quasi-pair-basis, and we prove that if a topological space X has a complete quasi-pair-base then X is domain representable

Noetherian π\pi-bases and Telg\'arsky's Conjecture

We investigate Noetherian families. By using a special Noetherian $\pi$-base, we give a result which states that existence of NONEMPTY's 2-tactic in the Banach-Mazur game on a space $X$, BM(X), if NONEMPTY has a winning strategy in $BM(X)$ and $X$ has the special Noetherian $\pi$-base. This result includes one of the Galvin's theorems which is important in this topic. From this result, we prove that for any topological space $X$ if $\pi w(X)\leq \omega_1$ and NONEMPTY has a winning strategy in $BM(X)$, then NONEMPTY has a 2-tactic in $BM(X)$. As a result of this fact, under $\mathsf{CH}$, we show that for any separable $T_3$ space $X$ if NONEMPTY has a winning strategy in $BM(X)$, then NONEMPTY has a 2-tactic in $BM(X)$. So, we prove that Telg{\'a}rsky's conjecture cannot be proven true in the realm of separable $T_3$ spaces, and more generally, in the class of spaces $X$ with $\pi w(X)\leq \omega_1$. We pose some questions about this topic

Some cardinal invariants on the space Cα(X,Y)

AbstractLet Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X such that closed under finite unions. We define two properties (E1) and (E2) on the triple (α,X,Y) which yield new equalities and inequalities between some cardinal invariants on Cα(X,Y) and some cardinal invariants on the spaces X, Y such as: TheoremIf Y is an equiconnected space with a base consisting of φ-convex sets, then for each f∈C(X,Y), χ(f,Cα(X,Y))=αa(X).we(f(X)).CorollaryLet Y be a noncompact metric space and let the triple (α,X,Y) satisfy (E1). The following are equivalent: (i)Cα(X,Y) is a first-countable space.(ii)π-character of the space Cα(X,Y) is countable.(iii)Cα(X,Y) is of pointwise countable type.(iv)There exists a compact subset K of Cα(X,Y) such that π-character of K in the space Cα(X,Y) is countable.(v)αa(X)⩽ℵ0.(vi)Cα(X,Y) is metrizable.(vii)Cα(X,Y) is a q-space.(viii)There exists a sequence {On:n∈ω} of nonempty open subset of Cα(X,Y) such that each sequence {gn:n∈ω} with gn∈On for each n∈ω, has a cluster point in Cα(X,Y)

Nuclear Köthe quotients of Frecht spaces.

Ph.D. - Doctoral Progra

Power stabılıty of k-spaces and compactness

It is proved that a topological space X is compact if X(m) is a k-space for each cardinal number m

An answer to a conjecture multiplicative maps on C(X, I)

An answer to the conjecture in [1] is given

Some upper bounds for density of function spaces

Let C-alpha(X, Y) be the set of all continuous functions from X to Y endowed with the set-open topology where alpha is a hereditarily closed, compact network on X which is closed Under finite unions. We proved that the density of the space C-alpha(X, Y) is at most iw(X) . d(Y) where iw(X) denotes the i-weight of the Tychonoff space X, and d(Y) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function psi, and Y has a base consists of psi-convex Subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal kappa, there is a pathwise connected space Y Such that pi-weight of Y is kappa, but Souslin number of the Space C-kappa(vertical bar 0, 1 vertical bar, Y) is 2(kappa)

The Banach-Stone theorem revisited

Let X and Y be compact Hausclorff spaces, and E and F be locally solid Riesz spaces. If pi : C(X. E) -> C(Y, F) is a 1-biseparating Riesz isomorphism then X and Y are homeomorphic, and E and F are Riesz isomorphic. This generalizes the main results of [Z. Ercan, S. Onal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829] and [X. Miao, C. Xinhe, H. Jiling, Banach-Stone theorems and Riesz algebras, J. Math. Anal. Appl. 313 (1) (2006) 177-183], and answers a conjecture in [Z. Ercan, S. Onal, Banach-Stone theorem for Banach lattice valued continuous functions. Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829]