Separation axiom

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff. The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties. The precise definitions of the separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition. Before we define the separation axioms themselves, we give concrete meaning to the concept of separated sets (and points) in topological spaces. (Separated sets are not the same as separated spaces, defined in the next section.) The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways). The separation axioms all say, in one way or another, that points or sets that are distinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense. Let X be a topological space. Then two points x and y in X are topologically distinguishable if they do not have exactly the same neighbourhoods (or equivalently the same open neighbourhoods); that is, at least one of them has a neighbourhood that is not a neighbourhood of the other (or equivalently there is an open set that one point belongs to but the other point does not).