Topology: the Journey into Separation Axioms

In this journey, we are going to explore the so called “separation axioms” in greater detail. We shall try to understand how these axioms are affected on going to subspaces, taking products, and looking at small open neighbourhoods. 1. What this journey entails 1.1. Prerequisites. Familiarity with definitions of these basic terms is expected: • Topological space • Open sets, closed sets, and limit points • Basis and subbasis • Continuous function and homeomorphism • Product topology • Subspace topology The target audience for this article are students doing a first course in topology. 1.2. The explicit promise. At the end of this journey, the learner should be able to: • Define the following: T0, T1, T2 (Hausdorff), T3 (regular), T4 (normal) • Understand how these properties are affected on taking subspaces, products and other similar constructions 2. What are separation axioms? 2.1. The idea behind separation. The defining attributes of a topological space (in terms of a family of open subsets) does little to guarantee that the points in the topological space are somehow distinct or far apart. The topological spaces that we would like to study, on the other hand, usually have these two features: • Any two points can be separated, that is, they are, to an extent, far apart. • Every point can be approached very closely from other points, and if we take a sequence of points, they will usually get to cluster around a point. On the face of it, the above two statements do not seem to reconcile each other. In fact, topological spaces become interesting precisely because they are nice in both the above ways. The first of these is related to the concern of “separation axioms”, and this is what we will look at here. 2.2. The concept of quasiorder. Definition. A quasiorder(defined) or preorder(defined) is a reflexive and transitive relation on a set. Under a quasiorder ≤ , two elements a and b are said to be equivalent(defined) if a ≤ b and b ≤ a. The above notion of equivalence partitions the set into equivalence classes, and the quasiorder boils down to a partial order (that is, a reflexive antisymmetric transitive relation) on the equivalence classes. Here are some examples of quasiorders. Suppose we have a collection of people. We say that a person a is not older than a person b if the age of a is less than or equal to the age of b (the age is simply counted in years, with no fractions). Clearly, this relation is reflexive: each person is not older than himself or herself. Moreover, it is also transitive: if a is not older than b, and b is not older than c, then the age of a is less than or equal to the age of c. Under this quasiorder, if a ≤ b and b ≤ a, we cannot conclude that a and b are the same person. What we can conclude is that they have the same age. In fact, “having the same age” is precisely the equivalence relation for the quasiorder, and each equivalence class is represented by a value of age. c ©Vipul Naik, B.Sc. (Hons) Math, 2nd Year, Chennai Mathematical Institute.