Strengthening connected Tychonoff topologies

Dimitri Shakhmatov, Mikhail Tkachenko, Vladimir V. Tkachuk, Richard G. Wilson


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 group.


Connected; Strongly σ-discrete; Submetrizable; Regular open set; Dense subset; Topological group; Quotient group; Free topological group

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