--- id: f4ac333d-2050-4f7c-b6b2-2f4b547ecb9a title: Menger Space in Topology abstract: "A Menger space is a topological space defined by a specific selection principle\ \ that generalizes the concept of \u03C3-compactness. It is characterized by the\ \ property that for every sequence of open covers, there exist finite subsets whose\ \ union covers the space. This concept is fundamental in general topology and set-theoretic\ \ topology." classification: primary: '515' secondary: - '51' - '514' udc_main_class: '5' tags: - topology - Menger space - mathematics - open covers - selection principle - general topology topics: - Mathematics - Topology - Pure Mathematics author: '' created_at: '2026-06-01T00:25:51.733916' updated_at: '2026-06-01T02:15:39.864167' sources: - type: url uri: https://en.wikipedia.org/wiki/Menger_space format: url_fetch udc_label: Topology version: '1' --- ## Card: Menger Space in Topology A Menger space is a topological space defined by a specific selection principle that generalizes the concept of σ-compactness. It is characterized by the property that for every sequence of open covers, there exist finite subsets whose union covers the space. This concept is fundamental in general topology and set-theoretic topology. ## Classification Primary: 515 | Secondary: 51, 514 | Tags: topology, Menger space, mathematics, open covers, selection principle, general topology | Topics: Mathematics, Topology, Pure Mathematics ## Content Menger space - Wikipedia Jump to content Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search Appearance Donate Create account Log in Personal tools Donate Create account Log in Contents move to sidebar hide (Top) 1 History 2 Menger's conjecture 3 Combinatorial characterization 4 Properties 5 References Toggle the table of contents Menger space Add languages Add links Article Talk English Read Edit View history Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Print/export Download as PDF Printable version In other projects Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia This article needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed. Find sources:   "Menger space"  –  news   · newspapers   · books   · scholar   · JSTOR ( August 2016 ) ( Learn how and when to remove this message ) In mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness . A Menger space is a space in which for every sequence of open covers U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of the space there are finite sets F 1 ⊂ U 1 , F 2 ⊂ U 2 , … {\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subset {\mathcal {U}}_{2},\ldots } such that the family F 1 ∪ F 2 ∪ ⋯ {\displaystyle {\mathcal {F}}_{1}\cup {\mathcal {F}}_{2}\cup \cdots } covers the space. History [ edit ] In 1924, Karl Menger [ 1 ] introduced the following basis property for metric spaces: Every basis of the topology contains a countable family of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz [ 2 ] observed that Menger's basis property can be reformulated to the above form using sequences of open covers. Menger's conjecture [ edit ] Menger conjectured that in ZFC every Menger metric space is σ-compact. A. W. Miller and D. H. Fremlin [ 3 ] proved that Menger's conjecture is false, by showing that there is, in ZFC, a set of real numbers that is Menger but not σ-compact. The Fremlin-Miller proof was dichotomic, and the set witnessing the failure of the conjecture heavily depends on whether a certain (undecidable) axiom holds or not. Bartoszyński and Tsaban [ 4 ] gave a uniform ZFC example of a Menger subset of the real line that is not σ-compact. Combinatorial characterization [ edit ] For subsets of the real line, the Menger property can be characterized using continuous functions into the Baire space N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} . For functions f , g ∈ N N {\displaystyle f,g\in \mathbb {N} ^{\mathbb {N} }} , write f ≤ ∗ g {\displaystyle f\leq ^{*}g} if f ( n ) ≤ g ( n ) {\displaystyle f(n)\leq g(n)} for all but finitely many natural numbers n {\displaystyle n} . A subset A {\displaystyle A} of N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} is dominating if for each function f ∈ N N {\displaystyle f\in \mathbb {N} ^{\mathbb {N} }} there is a function g ∈ A {\displaystyle g\in A} such that f ≤ ∗ g {\displaystyle f\leq ^{*}g} . Hurewicz proved that a subset of the real line is Menger iff every continuous image of that space into the Baire space is not dominating. In particular, every subset of the real line of cardinality less than the dominating number d {\displaystyle {\mathfrak {d}}} is Menger. The cardinality of Bartoszyński and Tsaban's counter-example to Menger's conjecture is d {\displaystyle {\mathfrak {d}}} . Properties [ edit ] Every compact, and even σ-compact, space is Menger. Every Menger space is a Lindelöf space Continuous image of a Menger space is Menger The Menger property is closed under taking F σ {\displaystyle F_{\sigma }} subsets Menger's property characterizes filters whose Mathias forcing notion does not add dominating functions. [ 5 ] References [ edit ] ^ Menger, Karl (1924). "Einige Überdeckungssätze der Punktmengenlehre". Selecta Mathematica . Sitzungsberichte der Wiener Akademie. Vol. 133. pp.  421– 444. doi : 10.1007/978-3-7091-6110-4_14 . ISBN   978-3-7091-7282-7 . {{ cite book }} : ISBN / Date incompatibility ( help ) ^ Hurewicz, Witold (1926). "Über eine verallgemeinerung des Borelschen Theorems". Mathematische Zeitschrift . 24 (1): 401– 421. doi : 10.1007/bf01216792 . S2CID   119867793 . ^ Fremlin, David; Miller, Arnold (1988). "On some properties of Hurewicz, Menger and Rothberger" (PDF) . Fundamenta Mathematicae . 129 : 17– 33. doi : 10.4064/fm-129-1-17-33 . ^ Bartoszyński, Tomek; Tsaban, Boaz (2006). "Hereditary topological diagonalizations and the Menger–Hurewicz Conjectures" . Proceedings of the American Mathematical Society . 134 (2): 605– 615. arXiv : math/0208224 . doi : 10.1090/s0002-9939-05-07997-9 . S2CID   9931601 . ^ Chodounský, David; Repovš, Dušan; Zdomskyy, Lyubomyr (2015-12-01). "Mathias Forcing and Combinatorial Covering Properties of Filters" . The Journal of Symbolic Logic . 80 (4): 1398– 1410. arXiv : 1401.2283 . doi : 10.1017/jsl.2014.73 . ISSN   0022-4812 . S2CID   15867466 . v t e Topology Fields General/Point-set set-theoretic continuum Pointless Algebraic combinatorial homology cohomology homotopy Differential Geometric low-dimensional knot Digital Key concepts Open set  /  Closed set Interior Continuity Space compact connected Hausdorff metric uniform second-countable Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism Metrics and properties Euler characteristic Betti number Winding number Chern number Orientability Key results Banach fixed-point theorem De Rham cohomology Invariance of domain Poincaré conjecture Tychonoff's theorem Urysohn's lemma Category Mathematics portal Wikibook Wikiversity Topics general algebraic geometric Publications Retrieved from " https://en.wikipedia.org/w/index.php?title=Menger_space&oldid=1353669163 " Category : Properties of topological spaces Hidden categories: CS1 errors: ISBN date Articles needing additional references from August 2016 All articles needing additional references This page was last edited on 11 May 2026, at 17:30  (UTC) . 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