1
$\begingroup$

Let $(X,T)$ be the subspace of $\Bbb R^2$ consisting of the points in the line segments joining $<0,1>$ to $<0,0>$ and to all the points $<\frac{1}{n}, 0>, n = 1, 2, ...$. Show that $(X,T)$ is connected but not locally connected.

I understand that $(X,T)$ is connected because it's path-connected in $\Bbb R^2$, but I'm having trouble seeing how it's not locally connected.

$\endgroup$
4
$\begingroup$

Hint: What does a small neighborhood of a point in the middle of the vertical segment look like?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.