The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

Zero-selectors and GO spaces

We are dealing with Vietoris continuous zero-selectors, i.e., they choose for each non-empty closed set F an isolated point in F. We show that the presence of a continuous zero-selector even on a small class of non-empty closed sets of a space X implies that X is scattered if X is metrizable or non-Archimedean or a P-space. Finally, using continuous zero-selectors, we characterize suborderable spaces which are subspaces of ordinals.     

Selectors in non-Archimedean spaces

Continuous selectors on the hyperspace F(X) are studied, when X is a non-Archimedean space. It is shown that a non-Archimedean space has a continuous selector if and only if it is topologically well orderable. Another characterization is given in terms of density and complete metrizability.     

Cut points in some connected topological spaces

We prove that a connected topological space with endpoints has exactly two non-cut points and every cut point is a strong cut point; it follows that such a space is a COTS and the only two non-cut points turn out to be endpoints (in each of the two orders) of the COTS. A non-indiscrete connected topological space with exactly two non-cut points and having only finitely many closed points is proved homeomorphic to a finite subspace of the Khalimsky line. Further, it is shown, without assuming any separation axiom, that in a connected and locally connected topological space X, for a, b in X, S[a,b] is compact whenever it is closed. Using this result we show that an H(i) connected and locally connected topological space with exactly two non-cut points is a compact COTS with end points.     

Characterizations of intervals via continuous selections

We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.     

A study of cut points in connected topological spaces

We prove that every H(i) subset H of a connected space X such that there is no proper connected subset of X containing H, contains at least two non-cut points of X. This is used to prove that a connected space X is a COTS with endpoints if and only if X has at most two non-cut points and has an H(i) subset H such that there is no proper connected subset of X containing H. Also we obtain some other characterizations of COTS with endpoints and some characterizations of the closed unit interval.     

The D-property of some Lindelöf spaces and related conclusions

It is shown that the space Cp(τω) is a D-space for any ordinal number τ, where . This conclusion gives a positive answer to R.Z. Buzyakova's question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshina's conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Resolvability of spaces having small spread or extent

In a recent paper O. Pavlov proved the following two interesting resolvability results:

(1)

If a T₁-space X satisfies Δ(X)>ps(X) then X is maximally resolvable.

(2)

If a T₃-space X satisfies Δ(X)>pe(X) then X is ω-resolvable.

Here ps(X) (pe(X)) denotes the smallest successor cardinal such that X has no discrete (closed discrete) subset of that size and Δ(X) is the smallest cardinality of a non-empty open set in X.In this note we improve (1) by showing that Δ(X)>ps(X) can be relaxed to Δ(X)?ps(X), actually for an arbitrary topological space X. In particular, if X is any space of countable spread with Δ(X)>ω then X is maximally resolvable.The question if an analogous improvement of (2) is valid remains open, but we present a proof of (2) that is simpler than Pavlov's.     

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X₁, X₂ such that Int X₁ is locally compact, no points of X₂ has a closed realcompact neighborhood, and $\text{Int(X}{}_1\text{?X}{}_2\text{)=?}$. It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.     

Subcompactness and domain representability in GO-spaces on sets of real numbers

In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R, then (R,τ) is subcompact, and so is any Gδ-subspace of (R,τ). We also show that if (X,τ) is a subcompact GO-space constructed on a subset X⊆R, then X is a Gδ-subset of any space (R,σ) where σ is any GO-topology on R with τ=σX_|. It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to Gδ-subsets. In addition, it follows that if (X,τ) is a subcompact GO-space constructed on any set of real numbers and if τS is the topology obtained from τ by isolating all points of a set S⊆X, then (X,τS) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known.We use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R. For example, examples show that there are subcompact GO-spaces constructed on subsets X⊆R where X is not a Gδ-subset of the usual real line. However, if (X,τ) is a dense-in-itself GO-space constructed on some X⊆R and if (X,τ) is subcompact (or more generally domain-representable), then (X,τ) contains a dense subspace Y that is a Gδ-subspace of the usual real line. It follows that (Y,τY_|) is a dense subcompact subspace of (X,τ). Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of (X,τ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: “Is there a domain-representable GO-space constructed on a subset of R that is not subcompact”? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.     

On nagata's problem for paracompact σ-metric spaces

Let (A) be the characterization of dimension as follows: Ind X?n if and only if X has a σ-closure-preserving base $\text{W}$ such that Ind B(W)?n?1 for every W?$\text{W}$. The validity of (A) is proved for spaces X such that(i) X is a paracompact σ-metric space with a scale {X_i} such that each X_i has a uniformly approaching anti-cover, or(ii) X is a subspace of the product ΠX_i of countably many L-spaces X_i, the notion of which is due to K. Nagami.(i) and (ii) are the partial answers to Nagata's problem wheter (A) holds or not for every M₁-space X.     

Zero-dimensional proximities and zero-dimensional compactifications

We introduce zero-dimensional proximities and show that the poset 〈Z(X),?〉 of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset 〈Π(X),?〉 of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that 〈Π(X),?〉 is isomorphic to the poset 〈B(X),⊆〉 of Boolean bases of X, and derive Dwinger's theorem that 〈Z(X),?〉 is isomorphic to 〈B(X),⊆〉 as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone-?ech compactification of X is a unique up to equivalence extremally disconnected compactification of X.     

Submaximal and door compactifications

In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:

(i)

D is co-finite in K(X);

(ii)

for each x∈K(X)?D, {x} is closed.

If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.     

A classification of CO spaces which are continuous images of compact ordered spaces

A topological space X is called a CO space, if every closed subset of X is homeomorphic to some clopen subset of X. Every ordinal with its order topology is a CO space. This work gives a complete classification of CO spaces which are continuous images of compact ordered spaces.     

On metric spaces with the properties of de Groot and Nagata in dimension one

A metric space (X,d) has the de Groot property GPn if for any points x₀,x₁,…,xn+2∈X there are positive indices i,j,k?n+2 such that i≠j and d(xi,xj)?d(x₀,xk). If, in addition, k∈{i,j} then X is said to have the Nagata property NPn. It is known that a compact metrizable space X has dimension dim(X)?n iff X has an admissible GPn-metric iff X has an admissible NPn-metric.We prove that an embedding f:(0,1)→X of the interval (0,1)⊂R into a locally connected metric space X with property GP₁ (resp. NP₁) is open, provided f is an isometric embedding (resp. f has distortion Dist(f)=‖fLip_‖⋅‖f−1Lip_‖<2). This implies that the Euclidean metric cannot be extended from the interval [−1,1] to an admissible GP₁-metric on the triode T=[−1,1]∪[0,i]. Another corollary says that a topologically homogeneous GP₁-space cannot contain an isometric copy of the interval (0,1) and a topological copy of the triode T simultaneously. Also we prove that a GP₁-metric space X containing an isometric copy of each compact NP₁-metric space has density ?c.     

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ<Δ(X), where Δ(X) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.