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As the tite explains, how to prove that a arcwised connected space is connected space?

Thanks for your help.

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Is arc-wise the same as path-connected? –  Matt N. Apr 22 '13 at 8:22
    
I don't know;(( –  Paul Apr 22 '13 at 8:24
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@MattN. Some authors make a subtle distinction: A path is a continuous map from the unit interval to the space and an arc is required to be a homeomorphism onto its image. Path-connected and arc-connected spaces are defined accordingly. For $T_2$-spaces these definitions are equivalent (one direction uses the Hahn-Mazurkiewicz theorem). In general they are distinct: A segment with two origins is an example of a path-connected space which is not arc-connected: there's no arc from one origin to the other. –  Martin Apr 22 '13 at 9:10
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up vote 3 down vote accepted

let $X$ be a pathconnected space. Suppose $X$ is not connected. Then $X=X_1\sqcup X_2$ where $X_1$ and $X_2$ are non empty disjoint open subsets of $X$. Now take a point $x_1\in X_1$ and $x_2\in X_2$ then there is no way to join these two points by a continuous path, which contradicts the pathconnectedness hyposthesis of $X$.

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