Strengthening connected Tychonoff topologies

Abstract

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.

Keywords

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