Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.4495

Some Generalizations of Fedorchuk Duality Theorem -- I

Georgi Dobromirov Dimov
View original source ↗Read source document ↗Export metadata ↓

Abstract

Generalizing Duality Theorem of V. V. Fedorchuk, we prove Stone-type duality theorems for the following four categories: all of them have as objects the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of all compact Hausdorff spaces and all open maps between them is proved. We also obtain equivalence theorems for these four categories. The versions of these theorems for the full subcategories of these categories having as objects all locally compact connected Hausdorff spaces are formulated as well.

Explore related subjects

math.GNmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Georgi Dobromirov Dimov. 2007-09-27. Some Generalizations of Fedorchuk Duality Theorem -- I. https://arxiv.org/abs/0709.4495

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$PF$-well-filtered spaces

This paper solves two open problems concerning the well-filteredness of topological spaces. The first, due to Shen, Xi and Zhao\cite{shen2023further}, asks whether a space must be open well-filtered whenever its upper space is so; the answer is negative; the Hofmann-Lawson space provides a counterexample. The second, posed by Li, Yuan and Zhao\cite{li2020unified}, inquires whether $PF$-well-filteredness and $PF$- sobriety are equivalent under the assumption of local compactness; we answer it in the affirmative. Subsequently, we give characterizations of $PF$-sobriety and $PF$-well-filteredness for $T_0$ spaces, respectively. On the basis of the characterizations, we obtain the following result: $PF$-sobriety and $PF$-well-filteredness coincide on the Rudin spaces, which extends the previously established equivalence for locally compact spaces, as every locally compact space is a Rudin space. Furthermore, we give a counterexample to show that for nonempty $T_0$ topological spaces $X$ and $Y$, if $Y$ is a $PF$-well-filtered ($PF$-sober) space, then the function space consisting of all continuous maps from $X$ to $Y$ equipped with the Isbell topology may not be a $PF$-well-filtered ($PF$-sober) space even when $X$ is a core-compact space. We also prove that if the function space comprising all continuous maps from nonempty $X$ to $Y$ endowed with the Isbell topology is $PF$-well-filtered ($PF$-sober), then $Y$ is also $PF$-well-filtered ($PF$-sober) even though $PF$-sobriety and $PF$-well-filteredness are not preserved under retracts. Finally, we show that the reflective hull of the category of $PF$-well-filtered ($PF$-sober) spaces with continuous maps in $TOP_0$ is $TOP_0$.

math.GN↗
arXiv2026

A non-sober core-compact meet-continuous dcpo

We construct a meet-continuous dcpo whose Scott space is core-compact but neither sober nor locally compact. The example is a proper sub-dcpo of a locally compact sober complete lattice, and restriction induces an isomorphism between their lattices of Scott-closed sets. It follows that a locally compact sober dcpo need not be determined, among all dcpos, by its Scott-closed-set lattice.

math.GN↗
arXiv2026

On identities in connected topological groups

We construct a connected (indeed, path-connected) Hausdorff topological group G such that, for an odd n >= 665, the identity x^n = e holds in some neighbourhood of the identity but does not hold in the whole group. This answers in the negative a question of Mycielski and Aronszajn (Question 2.48 of the Kourovka Notebook). The construction uses the Graev extension of metrics to free topological groups and results of Adian and Adian-Atabekyan (for n > 10^10, alternatively, of Ol'shanskii and Ashmanov-Ol'shanskii) on free Burnside groups and their central extensions.

math.GN↗