# Disjoint Union Topology - understanding operations on it

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

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Great questions Ernie! I had the same ones. – Andrew Sep 5 '12 at 0:34

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 $M$.

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Thank you, that's very helpful. If two sets in P and M have a union that is open in X, and if both sets have boundaries in their respective subspaces, would the boundary of their union in X be the union of their individual boundaries in P and M? – ernie Apr 10 '12 at 21:30
@ernie Yes. If $S\subset P$ and $T\subset M$, then the boundary of $S\cup T$ will be the union of the boundaries of $S$ and $T$. – Jim Belk Apr 10 '12 at 22:56

In this case both $P$ and $M$ are clopen (closed and open). So in particular the boundary of $P$ and $M$ is empty and no element of $P$ is a limit point of $M$ and vice-versa. On the other hand, an arbitrary subset $U$ of $X$ is open if and only if both $U\cap P$ and $U\cap M$ are open.

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Thank you. I assume then that if U is open in X its closure in X must be the union of the closure of U in P and the closure of U in M? – ernie Apr 10 '12 at 21:26
@ernie $cl(U)=cl(U\cap P)\cup cl(U\cap M)$ – azarel Apr 10 '12 at 22:44
@azarel : an arbitrary subset $U$ of $X$ is open if and only if both $U∩P$ and $U∩M$ are open. but one of these intersections must be empty, am i right? – palio Aug 26 '12 at 16:45
@palio No, both intersections can be non-empty. – azarel Aug 26 '12 at 18:04

Another very useful property of $P \sqcup M$ is that for any space $X$ and any continuous functions $p:P \to X$, $m: M \to X$ , the unique function $f: P \sqcup M \to X$ which agrees with $p,m$ on $P,M$ respectively, is continuous. Thus the disjoint union is good for constructing continuous functions from it, which is a kind of dual to the product, which is good for constructing functions $X \to P \times M$ in terms of its components.

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Thank you. I have seen that fact mentioned, but I have to confess I do not really understand the import of it. – ernie Apr 10 '12 at 21:32
A major import is that many topological constructions, such as this one, have a major role in constructing continuous functions either into or out of the space. The disjoint union is a special case of more general constructions, such as adjunction spaces, which are very useful in algebraic topology. – Ronnie Brown Apr 29 '12 at 11:35
I see. That's not where I'm trying to go with this at the moment, but useful to understand for the future. Thanks you. – ernie Apr 29 '12 at 15:34