I can find good explanations of how the disjoint union topology is constructed, but I am confused about how things such as complements, boundaries, limit points, etc. are to be understood in this context. For example, suppose we have two spaces, P and M and create their disjoint Union X with the disjoint union topology. It would seem that subsets of P and M must then be subsets of X that are disjoint. However, do they need to be separate as well or could a subset of P have limit points in a subset of M? With what open sets would the limit points be defined? How about the closure or boundary of unions of subsets of P and M? It seems from what I have been able to find that you could not define an open set in X that did not already exist in P or M, so I am confused. Any clarification or a pointer to a relevant treatment would be greatly appreciated.
Ernie
Best Answer
If $P$ and $M$ are disjoint topological spaces and $X = P\cup M$, then $X$ inherits a natural topology from $P$ and $M$, sometimes called the disjoint union topology. The open sets in this topology are all sets of the form $U\cup V$, where $U$ is open and $P$ and $V$ is open in $M$. In particular, since the empty set is open, any open subset of $P$ is open in $X$, and any open subset of $M$ is open in $X$.
The idea of this topology is that $P$ and $M$ form disconnected pieces of $X$, and do not interact in any way. Here are some basic properties:
No sequence in $P$ or subset of $P$ has a limit point in $M$, and vice-versa.
If $S\subset P$, then the closure of $S$ is also a subset of $P$. The same holds for $M$.