More on remainders close to metrizable spaces

AbstractThis article is a natural continuation of [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79–90]. As in [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79–90], we consider the following general question: when does a Tychonoff space X have a Hausdorff compactification with a remainder belonging to a given class of spaces? A famous classical result in this direction is the well known theorem of M. Henriksen and J. Isbell [M. Henriksen, J.R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958) 83–106].It is shown that if a non-locally compact topological group G has a compactification bG such that the remainder Y=bG∖G has a Gδ-diagonal, then both G and Y are separable and metrizable spaces (Theorem 5). Several corollaries are derived from this result, in particular, this one: If a compact Hausdorff space X is first countable at least at one point, and X can be represented as the union of two complementary dense subspaces Y and Z, each of which is homeomorphic to a topological group (not necessarily the same), then X is separable and metrizable (Theorem 12). It is observed that Theorem 5 does not extend to arbitrary paratopological groups. We also establish that if a topological group G has a remainder with a point-countable base, then either G is locally compact, or G is separable and metrizable

When a totally bounded group topology is the Bohr Topology of a LCA group

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of $w$. As an application, an example of a MAP group $(G,t)$ is given such that every closed, metrizable subgroup $N$ of $bG$ with $N \cap G = \{0\}$ preserves compactness but $(G,t)$ does not strongly respects compactness. Thereby, we respond to Questions 4.1 and 4.3 in [comftrigwu]

On topological groups with remainder of character k

[EN] In [A.V. Arhangel'skii and J. van Mill, On topological groups with a first-countable remainder, Top. Proc. 42 (2013), 157-163] it is proved that the character of a non-locally compact topological group with a first countable remainder doesn't exceed $\omega_1$ and a non-locally compact topological group of character $\omega_1$ having a compactification whose reminder is first countable is given. We generalize these results in the general case of an arbitrary infinite cardinal k. Bonanzinga, M.; Cuzzupè, MV. (2016). On topological groups with remainder of character k. Applied General Topology. 17(1):51-55. doi:10.4995/agt.2016.4376.SWORD5155171A. V. Arhangel'skii, Construction and classification of topological spaces and cardinal invariants, Uspehi Mat. Nauk. 33, no. 6 (1978), 29-84.A.V. Arhangel'skii, On the cardinality of bicompacta satisfying the first axiom of countability, Doklady Acad. Nauk SSSR 187 (1969), 967-970.A.V. Arhangel'skii and J. van Mill, On topological groups with a first-countable remainder, Topology Proc. 42 (2013), 157-163.R. Engelking, General Topology, Heldermann Verlag, Berlin, second ed., 1989.I. Juhász, Cardinal functions in topology--ten years later, Mathematical Centre Tract, vol. 123, Mathematical Centre, Amsterdam, 1980

Relative Collectionwise Normality

[EN] In this paper we study properties of relative collectionwise normality type based on relative properties of normality type introduced by Arhangel’skii and Genedi. Theorem Suppose Y is strongly regular in the space X. If Y is paracompact in X then Y is collectionwise normal in X. Example A T2 space X having a subspace which is 1− paracompact in X but not collectionwise normal in X. Theorem Suppose that Y is s- regular in the space X. If Y is metacompact in X and strongly collectionwise normal in X then Y is paracompact in X.Grabner, E.; Grabner, G.; Miyazaki, K.; Tartir, J. (2004). Relative Collectionwise Normality. Applied General Topology. 5(2):199-212. doi:10.4995/agt.2004.1970.SWORD19921252A. Arhangel'skii, "From classic topological invariants to relative topological properties", Scientiae Math. Japonicae 55 No 1 (2002), 153-201.A. Arhangel'skii and H Genedi, "Beginnings of the theory of relative topological properties", Gen. Top. Spaces and Mappings (MGU, Moscow, 1989), 3-48 (in Russian).A. Arhangel'skii and H Genedi, "Position of subspaces in spaces: relative versions of compactness, Lindel¨of properties, and separation axioms", Vestnik Moskovskogo Universiteta, Mathematika 44 No. 6 (1989), 67-69.A. Arhangel'skii and I. Gordienko, "Relative symmetrizability and metrizability", Comment. Math. Univ. Carol. 37 No 4 (1996), 757-774.R. Engelking, "General Topology" (PWN, Warsaw, 1977).I. Gordienko, "On Relative Properties of Paracompactness and Normality Type", Moscow Univ. Nath. Bul. 46 No. 1 (1991), 31-32.E. Grabner, G. Grabner and K. Miyazaki, " Properties of relative metacompactness and paracompactness type", Topology Proc. 25 (2000), 145-178.K. Miyazaki, "On relative paracompactness and characterizations of spaces by relative topological properties", Math. Japonica 50 (1999), 17-23.Smith, J. C., & Krajewski, L. L. (1971). Expandability and Collectionwise Normality. Transactions of the American Mathematical Society, 160, 437. doi:10.2307/1995819Y. Yasui, "Results on relatively countably paracompact spaces", Q and A in Gen. Top. 17 (1999), 165-174

Perfect mappings in topological groups, cross-complementary subsets and quotients

summary:The following general question is considered. Suppose that $G$ is a topological group, and $F$, $M$ are subspaces of $G$ such that $G=MF$. Under these general assumptions, how are the properties of $F$ and $M$ related to the properties of $G$? For example, it is observed that if $M$ is closed metrizable and $F$ is compact, then $G$ is a paracompact $p$-space. Furthermore, if $M$ is closed and first countable, $F$ is a first countable compactum, and $FM=G$, then $G$ is also metrizable. Several other results of this kind are obtained. An extensive use is made of the following old theorem of N. Bourbaki [5]: if $F$ is a compact subset of a topological group $G$, then the natural mapping of the product space $G\times F$ onto $G$, given by the product operation in $G$, is perfect (that is, closed continuous and the fibers are compact). This fact provides a basis for applications of the theory of perfect mappings to topological groups. Bourbaki's result is also generalized to the case of Lindelöf subspaces of topological groups; with this purpose the notion of a $G_\delta$-closed mapping is introduced. This leads to new results on topological groups which are $P$-spaces

Addition theorems and DD-spaces

summary:It is proved that if a regular space $X$ is the union of a finite family of metrizable subspaces then $X$ is a $D$-space in the sense of E. van Douwen. It follows that if a regular space $X$ of countable extent is the union of a finite collection of metrizable subspaces then $X$ is Lindelöf. The proofs are based on a principal result of this paper: every space with a point-countable base is a $D$-space. Some other new results on the properties of spaces which are unions of a finite collection of nice subspaces are obtained