Symmetrizable Space Research Articles

Classic Problems III

This is a survey of four problems that are “classics” in many different senses of the word, and of several related problems associated with each one. The numbering of the Classic Problems picks up where that of a similar article left off about four decades ago:IX.Is every point ofω⁎a butterfly point?X.Is there a nonmetrizable perfectly normal, locally connected continuum?XI.Is there a normal space with a σ-disjoint base that is not paracompact?XII.Is there a regular symmetrizable space with a non-Gδpoint?Several related problems are given for each classic problem. Consistency results are summarized, and there is a discussion of each problem that explains various implications among the related problems and justifies calling certain problems equivalent. For each classic problem, an appendix goes deeper into some implications and/or includes reminisces.There is a purely set-theoretic problem related to Classic Problem IX. Call a filter on a set D nowhere maximal if it does not trace an ultrafilter on any subset of D.Related Problem D. Is every free ultrafilter the join of two nowhere maximal filters?It is shown that the special case of ultrafilters on ω is actually equivalent to Classic Problem IX.

The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces

Abstract This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Frchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

Notes on Fréchet spaces

First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures. Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet. Finally, we study on a sufficient condition that a sequential space be Fréchet, a weakly first countable space be first countable, and a symmetrizable space be semi‐metrizable.

Final compactness and separability in regular symmetrizable spaces

In 1966 A. V. Arkhangel'skii posed the following question: Is it true that every regular finally compact symmetrizable space is separable? S. I. Nedev soon showed that a regular finally compact symmetrizable space is hereditarily finally compact. Consequently any counterexample to Arkhangel'skii's conjecture must be an L-space. Applying the technique of iterated forcing we prove that in the axiom systemZFC for set theory it is consistent to assume the existence of a regular (hereditarily) finally compact symmetrizable space X that is nonseparable. Thus it is impossible to prove using the axiom systemZFC that every regular finally compact symmetrizable space is separable. The space X has additional properties as well: it has a basis consisting of open/closed sets (i.e., it is zero-dimensional in the sense ofind, it can be mapped continuously and one-to-one onto a separable metric space, it is α-left and has cardinality ω1. Bibliography: 25 titles.

Compactifications of symmetrizable spaces

In response to questions of Arhangel'skil, we present examples of (1) (MA + -iCH) a symmetrizable space which is not metrizable but has a completely normal compactification and (2) (CH) a symmetrizable space which is not metrizable but has a perfectly normal compactification. In the construction of (2), a technique is developed which can be used to obtain first countable compactifications of many interesting examples.

Colorful Partitions of Cardinal Numbers

Use the two element subsets of κ, denoted by [κ]2, as the edge set for the complete graph on κ points. Write CP(κ, µ, v) if there is an edge coloring R: [κ]2 → µ with µ colors so that for every proper v element set X ⊊ κ, there is a point x ∈ κ ∼ X so that the edges between x and X receive at least the minimum of µ and v colors. Write CP⧣(K, µ, v) if the coloring is oneto- one on the edges between x and elements of X.Peter W. Harley III [5] introduced CP and proved that for κ ≧ ω, CP(κ+, κ, κ) holds to solve a topological problem, since the fact that CP(ℵ1, ℵ0, ℵ0) holds implies the existence of a symmetrizable space on ℵ1 points in which no point is a Gδ.

G σ-sets in symmetrizable and related spaces

In this paper we answer questions of Arhangel'skii and Michael by providing an example of a regular symmetrizable space which is not subparacompact and has a closed subset which is not a G σ-set. We also use the idea of sequential order to obtain some positive results, and several examples are provided which show that these results are in some sense the best possible.

Symmetrizable, -, and Weakly First Countable Spaces

A number of results are given concerning the character and cardinality of symmetrizable and related spaces. An example is given of a symmetrizable Hausdorff space containing a point that is not a regular Gδ , and a proof is given that if a point p of a symmetrizable Hausdorff space has a neighborhood base of cardinality , then p is a Gδ . It is shown that for each cardinal number m there exists a locally compact, pseudocompact, Hausdorff -space X with |X| ≧ m. Several questions of A. V. Arhangel'skii and E. Michael are partially answered.

Symmetrizable and related spaces

A study is made of a family of spaces which contains the symmetrizable spaces as well as many of the well-known examples of perfectly normal spaces.

Symmetrizable spaces and factor mappings

Main results: A pseudostratifiable symmetrizable space possesses a symmetric with weak Cauchy condition. A space is strongly symmetrizable if and only if it is a pseudo-open II-image of a metric space. There exists a zero-dimensional symmetrizable space without a symmetric with weak Cauchy condition.

A symmetrizable space that is not perfect

An example is given of a Hausdorff symmetrizable space which has a closed subset that is not a G δ {G_\delta } -subset (thus, it is not perfect) and which is not subparacompact. This example is then used in the construction of a symmetrizable T 1 {T_1} -space Y having a point x 0 {x_0} such that { x 0 } \{ {x_0}\} is not a G δ {G_\delta } -subset of Y.