# Do we always assume we are only dealing with one specific topology of a topological space X instead of all topologies?

I found the following definition of interior in Munkres's topology.

Given a subset $A$ of a topological space $X$, the interior of $A$ is defined as the union of all open sets contained in $A$.

Because $X$ could have multiple topologies, when it was talking about all open sets, does it mean the union of all open sets in all topologies, or the union of all open sets in only one topology?

More generally, when we are dealing with open sets of a topological space $X$, do we always tend to assume we are dealing with open sets in a specific topology instead of all possible topologies?

• You consider all open sets of a fixed topology. Notice that if we considered all possible topologies, then the interior of $A$ would be $A$ (thanks to the discrete topology), making this definition redundant. Nov 1, 2016 at 14:18

A topological space $X$ is really a pair $(X,\tau_X)$ of a set $X$ with a topology $\tau_X$ on $X$. By abuse of notation we use $X$ to refer to both the space and the underlying set. Hence, the open sets of $X$ are precisely the elements of $\tau_X$. The fact that the underlying set may have different topologies is almost never interesting or considered. In particular, it makes no sense to call a set open just because it is open in some topology that is not under consideration. Especially because that would just mean every set, since every set is open in some topology.