# Show that the real line with the lower limit topology and the Moore plane are not homeomorphic.

I found that the x-axis in the Moore plane is a closed and discrete set and the real line with the lower limit Topology don't have a set like that but I don't know how to conclude that there is not an homeomorphism between the Moore plane and the real line with the lower limit Topology. Thanks for any help.

Suppose $X$ is the Moore plane and $Y$ is the real line with the lower limit topology, and suppose by contradiction that $f:X\to Y$ is a homeomorphism.
Let $D\subseteq X$ be the real line in the Moore plane, which is a closed uncountable discrete subspace. Since $f$ is a homeomorphism, the restriction $g:D\to f(D)$ given by $g(d)=f(d)$ is also a homeomorphism. (Verify this.) Thus $f(D)$ is a closed uncountable discrete subset of $Y$. Since we know that $Y$ doesn't have a subspace like this (because, for example, it is Lindelöf), then we are done.