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connected; locally connected; free topological group; condensation; connectification
We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by ``nice'' continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather strong, but we show that it is difficult to construct spaces which would contain no infinite subspaces with a weaker connected $T_{3{1\over 2}}$-topology.
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