1
$\begingroup$

Consider in $\mathbb R$ the topology $\mathscr{T}_\ell$ having as basis: $$\mathscr{B}=\{[a, b): a, b\in\mathbb R, a<b\}.$$

Are the spaces $ [0, 1)$ and $(0, 1]$ homeomorphic when endowed with the subspace topology induced by $\mathscr{T}_\ell$?

Obs: I have already verified that $[0, 1)$ and $(0, 1]$ are neither connected nor compact with the above topology. Furtheremore, I guess the natural possible homeomorphism $f:[0, 1)\longrightarrow (0, 1]$, $x\mapsto f(x)=1-x$ is not continuous, so I'm a bit lost now.

$\endgroup$

1 Answer 1

1
$\begingroup$

Note that $1$ is an isolated point of $(0,1]$ with the subspace topology. On the other hand, there are no isolated points of $[0,1)$ with the subspace topology.

Since these subspaces differ on the topological property of having isolated points, they cannot be homeomorphic.

$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .