Posted by J. Holmer on November 30, 2007, 11:23 am, in reply to "Path Connectedness"
Board Administrator
Saying that something is a topological property means: If F:X \to Y is a homeomorphism and X possesses the property, then Y possesses the property.
Suppose X is p.c. and F is a homeomorphism. WTS Y is p.c. Let y_0, y_1 two points of Y. Since X is p.c. exists a path \gamma: [0,1] \to X such that \gamma(0)=F^{-1}(y_0) and \gamma(1)=F^{-1}(y_1). The desired path in Y is F(\gamma(t)).
Suppose X is connected and F is a homeomorphism. WTS Y is connected. Let U,V be a separating pair of open sets in Y. Then F^{-1}(U), F^{-1}(V) is a separating pair of open sets in X; contradiction.
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