Let $X$ be the ambient space, and $C \subset X$. In a subspace topology, $A$ is open in $C$ is equivalent to the existence of an open set $V\subset X$ such that $A=V\cap C$.
What are the definitions for interior, boundary, and closedness in $C$?(I've looked online, but I couldn't find anything…)
Do we still have
- $\text{int}_C(A) \cup \partial_C A = \text{cl}_C(A) $
- $\text{int}_C(A)=A$ equivalent to $A$ is open in $C$
- $A$ is closed in $C$ is equivalent to $A$ contains its boundary in $C$
?
Best Answer
The definitions of those terms in a space with the subspace topology are exactly the same as their definitions in any topological space. They are all given in terms of the open sets in the space (or in terms of other concepts that can be reduced to open sets); and the definition of the subspace topology tells you exactly what the open sets are. In particular, any theorem that is true in any topological space (I believe all three of your assertions are true theorems) is definitely true in a space with the subspace topology.
Sometimes we can be misled because sets that don't "look" open or closed really are in the subspace topology. For example, $[0,\infty)$ is a subspace of $\Bbb R$, and in that subspace the set $[0,1)$ is an open set; similarly, $\Bbb Z$ is a subspace of $\Bbb R$, and in that subspace every set is both open and closed. But that doesn't change the fact that these are real topological spaces with all of the properties that other topological spaces have.