Topological groups with dense compactly generated subgroups

[EN] A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) 0-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both 0-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both 0-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.Fujita, H.; Shakhmatov, D. (2002). Topological groups with dense compactly generated subgroups. Applied General Topology. 3(1):85-89. doi:10.4995/agt.2002.2115SWORD858931H. Fujita and D. B. Shakhmatov, A characterization of compactly generated metrizable groups, Proc. Amer. Math. Soc., to appear.B.A. Pasynkov, Almost-metrizable topological groups (in Russian), Dokl. Akad. Nauk SSSR 161 (1965), 281-284

Strengthening connected Tychonoff topologies

[EN] The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychonoff topology is considered. We show that every Tychonoff space X satisfying ω (X) ≤ c and c (X) ≤ N0 admits a finer strongly σ-discrete connected Tychonoff topology of weight 2c. We also prove that every connected Tychonoff space is an open continuous image of a connected strongly σ-discrete submetrizable Tychonoff space. The latter result is applied to represent every connected topological group as a quotient of a connected strongly σ-discrete submetrizable topological groupThe research was supported by Mexican National Council fo Science and Technology (CONACYT), grant number 400200-5-28411EShakhmatov, D.; Tkachenko, M.; Tkachuk, VV.; Wilson, RG. (2002). Strengthening connected Tychonoff topologies. Applied General Topology. 3(2):113-131. doi:10.4995/agt.2002.2058.SWORD1131313