Why is $[1,2]$ relatively open in $[1,2] cup [3,4]$?

From Tao, Analysis II.

Consider the set $X := [1,2] cup [3,4]$, with the usual metric. This set is disconnected because the sets $[1,2]$ and $[3,4]$ are open relative to $X$.

This question starts from the same book section but I need more clarification. I will try to explain my guess of what’s going on.

• A set $S$ is open iff it contains none of its boundary points.
• A boundary point is a point that is not an interior point or an exterior point.
• An interior point is a point where an open ball can be drawn around the point which is a subset of $S$.
• An exterior point is a point where an open ball can be drawn around the point which is disjoint from $S$.

For example, $1.5$ is an interior point of $[1,2]$ because letting $r=.1$, $(1.4,1.6) cap X = (1.4,1.6) subset [1,2]$.

Likewise, 1.0 is an interior point of $[1,2]$ because letting $r=.1$, $(0.9,1.1) cap X = (1.0,1.1) subset [1,2]$.

On the other hand, 3.5 is an exterior point because letting $r=.1$, $(3.4,3.6) cap X = emptyset$ .

All points in $X$ are either interior or exterior; X has no boundary points of $[1,2]$. So all of $[1,2]$‘s boundary points are outside $[1,2]$, a vacuous truth since there are no such points. So $[1,2]$ is relatively open in $X$.

Is that what’s happening, or is it something else?