What is a first countable space?

The definition of first countable space is standard and its meaning is very clear. But is that the case in the absence of the Axiom of Choice? The answer is negative because there are at least three choice-free versions of first countability. And, most likely, the usual definition does not correspond to what we want to be a first countable space. The three definitions as well as other characterizations of first countability are presented and it is discussed under which set-theoretic conditions they remain equivalent.     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

Disasters in topology without the axiom of choice

We show that some well known theorems in topology may not be true without the axiom of choice. Received: 29 August 1995 / Revised version: 23 June 2000 / Published online: 3 October 2001     

Monotone versions of countable paracompactness

One possible natural monotone version of countable paracompactness, MCP, turns out to have some interesting properties. We investigate various other possible monotone versions of countable paracompactness and how they are related.     

Embeddings of countable closed sets and reverse mathematics

Summary If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR ₀. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of .     

Kinna-Wagner selection principles, axioms of choice and multiple choice

We study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC).     

A normal screenable non-paracompact space

We construct a normal screenable non-paracompact spaces using ?⁺⁺, which is a consequence of V = L.     

Compact spaces, elementary submodels, and the countable chain condition, II

Given a space 〈X,T〉 in an elementary submodel of H(θ), define XM to be X∩M with the topology generated by . It is established that if XM is compact and satisfies the countable chain condition, while X is not scattered and has cardinality less than the first inaccessible cardinal, then X=XM. If the character of XM is a member of M, then “inaccessible” may be replaced by “1-extendible”.     

Dendrites with a countable closure of the set of end points

We investigate the problem of existence of universal elements in some families of dendrites with a countable closure of the set of end points. In particular, we prove that for each integer κ?3 and for each ordinal α?1 there exists a universal element in the family of all dendrites X such that ord(X)?κ and the α-derivative of the set clXE(X) contains at most one point.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

Solution of Shapirovskii's question

Boris Shapirovskii posed the following question: “Suppose a continuous mapping from one compact space onto another is given. Suppose that the π-character of any point in the domain is greater than the weight of the target space. Will there be two disjoint closed sets in the domain mapping onto?”. There is a zero-dimensional counterexample (Corollary 12), but the one cardinal up version is true in zero-dimensional case (Corollary 14). Nevertheless, there always will be two disjoint open sets mapping densely (Theorem 7).     

An explosion point space without the fixed point property

In 1988 A. Gutek proved that there exist one-point connectifications of hereditarily disconnected spaces that do not have the fixed point property. We improve on this result by constructing a one-point connectification of a totally disconnected space without the fixed point property.     

A first countable linearly Lindelöf not Lindelöf topological space

A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ℵω.Arhangel?skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed ℵ₀². Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵω>ℵ₀². They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵω. We answer this question in the negative by constructing a counterexample from MA+ℵω<ℵ₀².A modification of Alster?s Michael space that is first countable is presented.     

On order structure of the set of one-point Tychonoff extensions of a locally compact space

If a Tychonoff space X is dense in a Tychonoff space Y, then Y is called a Tychonoff extension of X. Two Tychonoff extensions Y₁ and Y₂ of X are said to be equivalent, if there exists a homeomorphism which keeps X pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of X. We identify those extensions of X which belong to the same equivalence classes. For two Tychonoff extensions Y₁ and Y₂ of X, we write Y₂?Y₁, if there exists a continuous function which keeps X pointwise fixed. This is a partial order on the set of all (equivalence classes of) Tychonoff extensions of X. If a Tychonoff extension Y of X is such that Y\X is a singleton, then Y is called a one-point extension of X. Let T(X) denote the set of all one-point extensions of X. Our purpose is to study the order structure of the partially ordered set (T(X),?). For a locally compact space X, we define an order-anti-isomorphism from T(X) onto the set of all nonempty closed subsets of βX\X. We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of X, the set of all one-point Lindelöf extensions of X, the set of all one-point pseudocompact extensions of X, and the set of all one-point ?ech-complete extensions of X, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of βX\X. We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of C^∗(X), partially ordered with inclusion, and the topology of subspaces of βX\X. We leave some problems open.     

Measurable cardinals and the cardinality of Lindelöf spaces

We obtain from the consistency of the existence of a measurable cardinal the consistency of “small” upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are Gδ sets.     

On countably compact,locally countable spaces

We study the cardinalities of countably compact, locally countableT ₃ spaces. For alln(<), there exists one of cardinality _n . IfV=L, then there exists one of cardinalityx iffx= orx =x. MA implies that there exists one of cardinality>2.     

Urysohn universal space, its development and Hausdorff's approach

Three approaches to a direct construction of Urysohn universal space are compared, namely those of Urysohn, Hausdorff and Katětov. More details are devoted to the unpublished Hausdorff's approach that is shown to work in a more general situation, too.     

On a core concept of Arhangel'ski?

Arhangel'ski? [A.V. Arhangel'ski?, Locally compact spaces of countable core and Alexandroff compactification, Topology Appl. 154 (2007) 625-634] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper.     

Nagata's conjecture and countably compact hulls in generic extensions

Nagata conjectured that every M-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. Although this conjecture was refuted by Burke and van Douwen, and A. Kato, independently, but we can show that there is a c.c.c. poset P of size ω² such that in VP Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space X∈V is an M-space in VP then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in VP). By a result of Morita, it is enough to show that every first countable regular space from the ground model has a first countable countably compact extension in VP. As a corollary, we also obtain that every first countable regular space from the ground model has a maximal first countable extension in model VP.