1. ## connectedness

Let X denote rational points of the interval [0,1]*0 of R^2.Let T denote the union of all line segments joining the point p=(0,1) to the points of X.
1)Show that T is path connected but is locally connected only at the point p.
2)find a subset of R^2 that is path connected but is locally connected at none of its points.

2. Originally Posted by math.dj
Let X denote rational points of the interval [0,1]*0 of R^2. Let T denote the union of all line segments joining the point p=(0,1) to the points of X.
1)Show that T is path connected but is locally connected only at the point p.
2)find a subset of R^2 that is path connected but is locally connected at none of its points.
For (2), let Y denote the set of rational points of the interval [–1,0]*1. Let S denote the union of all line segments joining the point q=(0,0) to the points of Y. Now with T as described in the question, take $\displaystyle R = S\cup T$. Then R is path-connected, because every point in R is on a path to the origin, either in a direct line (if the point is in S) or by going along a line to p and then down the y-axis to the origin (if the point is in T). But R is not locally connected anywhere.

3. X is cone of [0,1] and any cone is contractable. And contractable spaces are path connected.

4. I accept with information:Let S denote the union of all line segments joining the point q=(0,0) to the points of Y. Now with T as described in the question, take . Then R is path-connected, because every point in R is on a path to the origin, either in a direct line (if the point is in S) or by going along a line to p and then down the y-axis to the origin (if the point is in T).
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