Proposition

Give an example of a topological embedding that is neither an open map nor a closed map.

Solution

Let $A = [0, 1)$. Then the inclusion map $i_A: A \rightarrow \mathbb{R}$ is a topological embedding as shown here.

$A$ is both open and closed in $A$. However, $i_A(A) = A$ is neither open nor closed in $\mathbb{R}$. Therefore, $i_A$ is a topological embedding that is neither open nor closed.