There’s a whole list of properties of topological spaces that we may want to refer to called the separation axioms. Even when two points are distinct elements of the underlying set of a topological space, we may not be able to tell them apart with topological techniques. Points are separated if we can tell them apart in some way using the topology. Today we’ll discuss various properties of separation, and tomorrow we’ll list some of the more useful separation axioms we can ask that a space satisfy.
First, and weakest, we say that points and in a topological space are “topologically distinguishable” if they don’t have the same collection of neighborhoods — if . Now maybe one of the collections of neighborhoods strictly contains the other: . In this case, every neighborhood of is a neighborhood of . a forteriori it contains a neighborhood of , and thus contains itself. Thus the point is in the closure of the set . This is really close. The points are topologically distinguishable, but still a bit too close for comfort. So we define points to be “separated” if each has a neighborhood the other one doesn’t, or equivalently if neither is in the closure of the other. We can extend this to subsets larger than just points. We say that two subsets and are separated if neither one touches the closure of the other. That is, and .
We can go on and give stronger conditions, saying that two sets are “separated by neighborhoods” if they have disjoint neighborhoods. That is, there are neighborhoods and of and , respectively, and . Being a neighborhood here means that contains some open set which contains and contains some open set which contains
, and so the closure of is contained in the open set, and thus in . Similarly, the closure of must be contained in .. We see that the closure of is contained in the complement of , and similarly the closure of is in the complement of , so neither nor can touch the other’s closure. Stronger still is being “separated by closed neighborhoods”, which asks that and be disjoint closed neighborhoods. These keep and even further apart, since these neighborhoods themselves can’t touch each other’s closures.
The next step up is that sets be “separated by a function” if there is a continuous function so that for every point we have , and for every point we have . In this case we can take the closed interval whose preimage must be a closed neighborhood of by continuity. Similarly we can take the closed interval whose preimage is a closed neighborhood of . Since these preimages can’t touch each other, we have separated and by closed neighborhoods. Stronger still is that and are “precisely separated by a function”, which adds the requirement that only points from go to and only points from go to .
This list of separation