# General Topology

## New submissions

[ total of 6 entries: 1-6 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Products of $H$-separable spaces in the Laver model
Journal-ref: Topology Appl. 239 (2018), 115-119
Subjects: General Topology (math.GN); Logic (math.LO)

We prove that in the Laver model for the consistency of the Borel's conjecture, the product of any two $H$-separable spaces is $M$-separable.

[2]
Title: Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses
Subjects: General Topology (math.GN); Group Theory (math.GR)

Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$\mathcal I$ set in a Polish group. Here $\mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $B\subset X$ of a Polish group $X$ is called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that $f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically Haar-$\mathcal I$ if the set of witness functions $\{f\in C(K,X):\forall x\in X\;\;f^{-1}(B+x)\in\mathcal I\}$ is comeager in the function space $C(K,X)$. We study (generically) Haar-$\mathcal I$ sets in Polish groups for many concrete and abstract ideals $\mathcal I$, and construct the corresponding distinguishing examples. Also we establish various Steinhaus properties of the families of (generically) Haar-$\mathcal I$ sets in Polish groups for various ideals $\mathcal I$.

[3]
Title: Generalised Net Convergence Structures in Posets
Subjects: General Topology (math.GN)

In this paper, we introduce the notion of $\mathcal{M}$-convergence and $\mathcal{MN}$-convergence structures in posets, which, in some sense, generalise the well-known Scott-convergence and order-convergence structures. As results, we give a necessary and sufficient conditions for each generalised convergence structures being topological. These results then imply the following two well-established results: (1) The Scott-convergence structure in a poset $P$ is topological if and only if $P$ is continuous, and (2) The order-convergence structure in a poset $P$ is topological if and only if $P$ is $\mathcal{R}^*$-doubly continuous.

### Cross-lists for Tue, 20 Mar 18

[4]  arXiv:1803.06181 (cross-list from math.LO) [pdf, ps, other]
Title: Products of Lindelöf spaces with points $G_δ$
Authors: Toshimichi Usuba
Subjects: Logic (math.LO); General Topology (math.GN)

We show that if CH holds and either (i) there exists an $\omega_1$-Kurepa tree, or (ii) $\square(\omega_2)$ holds, then there are regular $T_1$ Lindel\"of spaces $X_0$ and $X_1$ with points $G_\delta$ such that $e(X_0 \times X_1)>2^\omega$.

### Replacements for Tue, 20 Mar 18

[5]  arXiv:1801.04176 (replaced) [pdf, ps, other]
Title: Non-meagre subgroups of reals disjoint with meagre sets
Authors: Ziemowit Kostana
Subjects: General Topology (math.GN)
[6]  arXiv:1803.01173 (replaced) [pdf, ps, other]
Title: Lattices of coarse structures
Comments: coarse structure, ballean, lattice of coarse structures
Subjects: General Topology (math.GN)
[ total of 6 entries: 1-6 ]
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