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?"
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.
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.
