Consider the subset A of the plane which consists of all poi

Consider the subset A of the plane which consists of all points (cost /t, sin t /t) for all t > 0 and the origin. Is A path-connected? Explain.

Solution

The points (cos(t)/t, sin(t)/t) will span from origin to the each and every point including when t is just greater than 0. as soon as t will increase since cos(t) and sin(t) can\'t be greater than 1, the points will be going to lie in a small region close to (0,0).

Since the relation is on R, it implies that it contains all set of points satisfying this relation => that path A is connected.