Some questions by Mikhail Matveev

Some questions

Mikhail Matveev

Department of Mathematics, University of California, Davis
Davis, CA 95616, USA (address valid till June 30, 2000)
e-mail: misha_matveev@hotmail.com

Most of these questions (at least most from the section on star covering properties) are from the author's survey on star covering properties published in Topology Atlas. For details or motivation see this survey

or contact the author by the e-mail address above. All spaces are assumed to be Tychonoff unless otherwise stated.

Pseudocompactness

A pseudocompact extension, pX, of a space X is a pseudocompact space that contains X as a dense subspace; pX is a minimal pseudocompact extension of X if no proper subspace of pX is a pseudocompact extension of X.

Question 1 Is there a space without a minimal pseudocompact extension?

Homogeneity

A space X is called B-homogeneous ("B"' is for "basically"') if there is a base for X such that every element of which can be mapped onto every other by an autohomeomorphism of X.

Question 2 Is every topological vector space B-homogeneous?

A separable space X is called CDH ("CDH"' is for "countably dense homogeneous'') if for every two dense countable subspaces, D₁ and D₂ of X there is an autohomeomorphism of X that maps D₁ onto D₂.

Question 3 Is it consistent that 2^\tau is CDH for some uncountable \tau?

Monotonicity

A space X is called monotonically compact (monotonically Lindelöf) provided there is a mapping that assigns to every open cover U of X a finite (resp. countable) open refinement R(U) such that R(U₁) refines R(U₂) as soon as U₁ refines U₂.

Question 4 Is every monotonically compact space metrizable?

Question 5 Is every countable monotonically Lindelöf space metrizable?

Inverse properties

Let A be a family of subsets of a set X. A partial inversement of A is a family { p(A) : A in A} such that p(A) is either A or X \ A for every A in A. A space X is called inversely compact (inversely Lindelöf) if every open cover of X has a partial inversement which contains a finite (resp. countable) subcover of X. In other words, a space X is inversely compact if every independent family of closed subsets of X has nonempty intersection.

Question 6 Is there a Hausdorff (regular, Tychonoff, normal) inversely compact space which is not compact?

Question 7 Is the discrete sum of two inversely Lindelöf spaces inversely Lindelöf?

Star covering properties

Compacness-type properties

If U is a cover of X and A is a subset of X then St(A, U) = St¹(A, U) = \cup { U in U : U \cap A is non-empty } and Stⁿ⁺¹(A, U) = St(Stⁿ(A, U), U) for n in \omega. A space X is n-starcompact if for every open cover of X the cover { Stⁿ({x}, U) : x in X } contains a finite subcover. A space X is n[1/2]-starcompact if for every open cover of X the cover { Stⁿ(U, U) : U in U } contains a finite subcover.

Question 8 Suppose X is a union of countably many dense countably compact subspaces. Must X be 1[1/2]-starcompact?

Question 9 Suppose X = Y \cup Z where Y is dense in X and countably compact and Y Lindelöf. Must X be 1[1/2]-starcompact?

Question 10 (M. Reed) Is every 1[1/2]-starcompact Moore space compact?

Question 11 (I. Tree) Is every pseudocompact CCC space 2-starcompact?

In particular:

Question 12 Must a pseudocompact space X be 2-starcompact provided it is one of the following:
- centered-Lindelöf,
- linked-Lindelöf,
- with \sigma-centered base,
- with \sigma-linked base?

Question 13 Does there exist a pseudocompact topological group which is not 2-starcompact?

Question 14 Does there exist a pseudocompact, not 2-starcompact (dense) subgroup in D^\tau?

A space X is mini-n-starcompact if it is n-starcompact and no infinite subspace of X is n-[1/2]-starcompact.

Question 15 Does there exist a regular mini-1[1/2]-starcompact space? A regular mini-2-starcompact space? How about topological groups?

A space X is called absolutely 2-starcompact provided for every open cover U and every dense subspace Y of X there is a finite subset A of Y such that St²(A, U) = X.

Question 16 Does there exist, within ZFC, a 2-starcompact space which is not absolutely 2-starcompact?

Question 17 Does there exist a first countable (or at least countably tight) 2-starcompact space which is not absolutely 2-starcompact?

Question 18 Is the product X x Y 2-starcompact provided X is 2-starcompact and Y is compact?

Question 19 Is there a cardinal \kappa such that 1[1/2]-starcompactness (2-starcompactness) of X^\kappa implies 1[1/2]-starcompactness (2-starcompactness) of X^\tau for all \tau?

Absolute countable compactness and property (a)

A space X has property (a) (or is an (a)-space) if for every open cover of X and every dense subspace Y of X there is a discrete, closed (in X) subset A of Y such that St(A, U) = X. A countably compact space with property (a) is called acc (absolutely countably compact).

Question 20 Characterize those X for which the product X x \beta X is an (a)-space (an acc space).

Question 21 Suppose X^\omega₁ is an (a)-space. Must X be countably compact?

Question 22 Must a pseudocompact (a) space be countably compact?

Question 23 Characterize the spaces all continuous images of which are (a)-spaces.

Question 24 Is there a Tychonoff space without a dense subspace which is an (a)-space? Is there a countably compact space without a dense acc subspace?

Star-Lindelöfness

The star-Lindelöf number of X, St-l(X), is the least cardinal \tau such that for every open cover U of X there is a subset A of X, of cardinality at most \tau, such that St(A, U) = X.

The linked Lindelöf number of X, ll(X), is the least cardinal \tau such every open cover of X has a subcover representable as the union of \tau many linked subfamilies.

Question 25 Is the inequality 2^|K| <= (\chi(X))^St-l(X) (or ll(X) or a(X) instead of St-l(X), or w(X) instead of \chi(X)) true for every closed discrete subset of a normal space X?

Question 26 Is there a ZFC example of a normal discretely star-Lindelöf space with uncountable extent?

Question 27 Can every locally compact (or every Cech-complete, or every realcompact) Tychonoff space be embedded into a Tychonoff star-Lindelöf space as a closed subspace?

Question 28 Does there exist a space which is both normal and discretely star-Lindelöf (or star-Lindelöf with property (a)) and has extent c?

Question 29 Can the extent of a star-Lindelöf (a)-space be greater than c?

Date published: June 6, 2000. Revised: June 14, 2000