Density character of subgroups of topological groups

Preprint English OPEN
Leiderman, Arkady; Morris, Sidney A.; Tkachenko, Mikhail G.;
(2015)
• Subject: Mathematics - General Topology | Primary 54D65, Secondary 22D05
arxiv: Mathematics::General Topology

A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective... View more
• References (26)
26 references, page 1 of 3

with δ = 1, 2 are disjoint. Let us put ti,δ = zi,δ(r) for all i = 1, . . . , k and δ = 1, 2.

Since the set S = {sj : j ∈ ω} is dense in Π, we can choose, for every i ≤ n and ǫ ǫ δ = 1, 2, an element si,δ ∈ S ∩ ti,δW ∩ W ti,δ. Then si,δ ∈ ti,δW for each ǫ = ±1.

Furthermore, according to our choice of the variables zi, the elements si,δ can be chosen to satisfy si,δ = sl,δ whenever di = dl, where i, l ≤ n and δ = 1, 2. It now follows from O1 ∩ O2 = ∅ that the elements (ii) there exist rational numbers 0 = b0 < b1 < · · · < bm−1 < bm = 1 such that f is constant on each subinterval [bk, bk+1) and f (bk) = gk ∈ D for k = 0, 1, . . . , m − 1.

for every r ∈ [0, 1]. Hence µ ({x ∈ J : Φ(r, x) ∈/ U }) < ǫ for each r ∈ [0, 1]. In other words, the path Φ lies in O(U, ε), so the set O(U, ε) is path-connected. Since the sets of the form G0 ∩ O(U, ε) constitute a base for G0 at the identity, this completes the proof of the theorem.

 Alexander V. Arhangel'skii and Mikhail G. Tkachenko, Topological Groups and Related Structures, Atlantis Series in Mathematics, Vol. I, Atlantis Press and World Scientific, ParisAmsterdam, 2008.

 Wistar W. Comfort, Topological Groups, Chapter 24 in Handbook of Set-Theoretic Topology, Kenneth Kunen and Jerry E. Vaughan, eds., North-Holland, Amsterdam, New York, Oxford, 1984.

 Wistar W. Comfort and Gerald L. Itzkowitz, Density Character in Topological Groups, Math. Ann. 226 (1977), 223-227.

 Richard Engelking, On the double circumference of Alexandroff, Bull. Acad. Pol. Sci. S´er. Math. 16 (1968), 629-634.

 Richard Engelking, Cartesian products and dyadic spaces, Fund. Math. 57, (1965), 287-304.

 Jorge Galindo,t Mikhail Tkachenko, Montserrat Bruguera, and Constancio Herna´ndez, Extensions of reflexive P -groups, Topol. Appl. 163 (2014), 112-127.

• Similar Research Results (5)
 Pro-Lie groups which are infinite-dimensional Lie groups (2006) 95% Pro-Lie Groups: A Survey with Open Problems (2015) 92% Lattices of homomorphisms and pro-Lie groups (2016) 89% Amenability and representation theory of pro-Lie groups (2016) 84% Coadjoint orbits in representation theory of pro-Lie groups (2017) 77%
• Metrics
Share - Bookmark