While revising IB Metric and Topological Spaces, I came across the following cool alternative to the topologists sine curve as an example of a space that is connected but not pathwise connected.

The idea is to build a really really long line (in fact we shall build a ray, but constructing a line can be achieved by an obvious gluing job). It will be connected for the same reasons as the real line, but not pathwise connected just because any path will be too short. To realise our goal, we pick the smallest uncountable ordinal \omega_1 and just take X=\omega_1 \times [0,1) \cup \{\infty\} with the obvious (order) topology generated by interpreting this as placing uncountably many intervals end-to-end (and \infty > (\alpha,x) for all (\alpha,x) \in \omega_1 \times [0,1)).

Consequently we get a really really long ray, which is far longer than any continuous image of a closed interval of real numbers, so it isn’t pathwise connected (any path from (0,0) to \infty must cross uncountably many intervals, but there is a rational number in the inverse image of each interval.

Advertisements