Let $X \subset \mathbb{R}$ with the standard topology.
Let $c \in \mathbb{R} \setminus \{0\}$, $X_{c} = \{cx | x \in X \}$.
What "topological properties" does $X_{c}$ inherit from $X$?
I was able to show that if $X$ is open, then $X_{c}$ is, and compactness is easy, as is connectedness. But, what about being meager, or with the property of Baire?
Honestly, I can't think of a topological property that won't be inherited.