Globalizing weak homotopy equivalences

We generalize the basic quasifibration theorem of Dold and Thom (1958), replacing quasifibrations by weak equivalences.     

Globalizing weak homotopy equivalences

We generalize the basic quasifibration theorem of Dold and Thom (1958), replacing quasifibrations by weak equivalences.     

Generalized homotopy property

Vietnam. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 198–201, January–February, 1990.     

On the stability of the weak Pinsker property

We say that a dynamical system has the weak Pinsker property when it possesses decreasing factors of arbitrarily small entropy such that every one of these factors splits off with a Bernoulli complement. We prove here that this property is stable under the taking of factors and ofd d limits.     

On the homotopy invariance of torsion for covering spaces

We prove the homotopy invariance of torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the cohomology of the covering space vanishes, the homotopy invariance was established by Lück. We also give some applications of our results.

     

Proof of semialgebraic covering mapping cylinder conjecture with semialgebraic covering homotopy theorem

In this paper we prove the semialgebraic version of Palais' covering homotopy theorem, and use this to prove Bredon's covering mapping cylinder conjecture positively in the semialgebraic category. Bredon's conjecture was originally stated in the topological category, and a topological version of our semialgebraic proof of the conjecture answers the original topological conjecture for topological G-spaces over “simplicial” mapping cylinders.     

On a strong property of the weak subalgebra lattice

In the present paper, we apply results from [Pió1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A. Received September 5, 1997; accepted in final form October 7, 1998.     

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.     

Functional homology operations and the weak homotopy type

Translated from Matematicheskie Zametki, Vol. 45, No. 5, pp. 76–86, May, 1989.     

Weak covering and the tree property

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if has the tree property then and is weakly compact in K. Received: 11 June 1997     

On a generalization of principal weak flatness property

In this article a generalization of principal weak flatness of acts is defined. Using this new property the characterization of some new classes of monoids are investigated. Furthermore, many known results are generalized.     

The covering homotopy extension problem for compact transformation groups

It is shown that the orbit space of universal (in the sense of Palais) G-spaces classifies G-spaces. Theorems on the extension of covering homotopy for G-spaces and on a homotopy representation of the isovariant category ISOV are proved.     

On a separable weak version of the bounded approximation property

Recently, Lee introduced and studied the separable weak bounded approximation property (BAP). Lee proved that the separable weak BAP of \({X^*}\), the dual space of a Banach space \({X}\), coincides with the BAP of \({X^*}\) whenever \({X^{**}}\) has the weak Radon–Nikodým property. We show that the separable weak BAP and the BAP are always the same properties.     

The diamond covering property axiom

The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.     

Some weak covering properties and infinite games

We show that (I) there is a Lindelöf space which is not weakly Menger, (II) there is a Menger space for which TWO does not have a winning strategy in the game G_fin(O,D_o). These affirmatively answer questions posed in Babinkostova, Pansera and Scheepers [Babinkostova L., Pansera B.A., Scheepers M., Weak covering properties and infinite games, Topology Appl., 2012, 159(17), 3644–3657]. The result (I) automatically gives an affirmative answer of Wingers’ problem [Wingers L., Box products and Hurewicz spaces, Topology Appl., 1995, 64(1), 9–21], too. The selection principle S₁(D_o,D_o) is also discussed in view of a special base. We show that every subspace of a hereditarily Lindelöf space with an ortho-base satisfies S₁(D_o,D_o).     

Weak separation axioms and weak covering properties

We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly^? subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that,1) on weakly subparacompact spaces, countable compactness = compactness, ω₁-compactness = Lindelöfness;2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and3) on weakly subparacompact regular T₁-spaces with rudimentary normality, collectionwise σ-normality = screenability = collectionwise normality = paracompactness.The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wicke?s factorization of paracompactness (over Hausdorff spaces) along with Krajewski?s are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them.