Do homeomophisms of subspaces preserve the subspaces topological structure in the parent space? To make this question more precise, suppose $X$ and $Y$ are topological spaces and let $A \subseteq X$ and $B \subseteq Y$ be subspaces. Suppose $A$ is open in $X$ and $A$ is homomorphic to $B$, does it follow that $B$ is open in $Y$?
What if I replaced "open", with "closed" or "compact" or "connected", or "locally connected" etc.
Loosely speaking this amounts to "Do homeomophisms of subspaces preserve the subspaces topological structure in the parent space?"
 A: For closed sets, the answer is negative: take $\mathbb Z$ and $\left\{\frac1n\,\middle|\,n\in\mathbb{N}\right\}$ as subsets of $\mathbb R$.
For open sets, the answer is negative too: take the subsets $(0,1]$ and $(1,2]$ of $(0,2]$. They are homeomorphic, but $(1,2]$ is open, whereas $(1,2]$ isn't.
For connected and compact sets, the answer is affirmative: being compact or connected does not depend upon the environment.
A: Homeomorphisms preserve topological properties, but relative properties need not be preserved.

Openness and closedness are relative properties, not topological properties.

For example . . .

Any open subset of $\mathbb{R}$ is homemeomorphic to itself, regarded as a subspace of $\mathbb{R^2}$, but in $\mathbb{R^2}$, it's no longer open.

The graph of $e^x$ is closed in $\mathbb{R^2}$, and is homeomorphic to the subspace $(0,\infty)$ of $\mathbb{R}$, which is not closed in $\mathbb{R}$.

However compactness, connectedness, local connectedness are topological properties, so are preserved by homeomorphisms.

The bottom line: If you can define a property using the topology of the space itself, without regard to a parent space, then it's a topological property.
