In this chapter, we shall study separation properties of topological spaces, vaguely speaking, whether two distinct points, a point and a closed set not containing the point, or two disjoint closed sets can be separated by disjoint open sets. On one extreme, we have indiscrete spaces in which nothing can be separated, and on the other end, we have discrete spaces in which every subset is open and it possesses all the separation properties. Metric spaces satisfy all these properties, and so, in a way, separation axioms attempt to seek how far is a topological space from being metrizable. Initially, the definition of topology given by Felix Hausdorff included a condition that we now know as the T2-axiom or the Hausdorff property. Later on, separation axioms were studied under the name accessible spaces, which was introduced by Fréchet, now known as T1-spaces and also as Fréchet space. The T0-space was introduced by Kolmogorov, and T0-spaces are also called Kolmogorov spaces. In 1923, Heinrich Tietze introduced the notation Ti for these spaces, which comes from the German word “Trennungsaxiomen,” which means “separation axioms,” and it has nothing to do with his name.

In this chapter, we introduce basic notions such as sets, functions, relations and order which are very fundamental and used, or assumed, later in the course of the book. Readers may already be familiar with most of these topics. Here we give a quick and brief overview of these concepts. The preliminaries covered in this chapter are to make the readers familiar with the notations and terminologies and for their quick handy reference, but these cannot replace a course in the theory of sets and functions.

1.1 Sets and Set Operations

Sets and elements

If an object x is in a set A, we say that “x belongs to A” or “x is an element of A” or “x is a member of A”, and we write x ∈ A. The negation of x ∈ A is written as x 6∈ A. There are two ways of writing a set. One is the list form, which is also called the roster form. For instances, {1, 2, 3, 4, 5, 6, 7, 8, 9} and {1, 2, . . . , 1000} are finite sets written in the roster form, while the set of natural numbers N = {1, 2, 3, . . .} and the set of even integers 2Z = {. . . , −4, −2, 0, 2, 4, . . .} are infinite sets written in the roster form. It is not always convenient or possible to write a set in the roster form by listing all its elements, for example, the set of real numbers. The other way of writing a set is the set-builder form, which is of the form {x : x satisfies property P}, where P is some property used to describe the set. For example, the set 2Z, of even integers we wrote above in the list form, can be written in the set-builder form as 2Z = {x | x is an even integer}, or using the set of integers in set builder form, it can be written as 2Z = {x | x = 2n for some integer n}. The same set in the set-builder form can be written using quantifiers as 2Z = {x | ∃ n ∈ Z such that x = 2n} or {x | ∃ n ∈ Z (x = 2n)}, where ∃ is the notation for the existential quantifier, which means “there is” or “there exists”.

In this chapter, we shall see how new topological spaces may be created from given topological space(s). We already saw one such technique Section 2.4 of Chapter 2, where, given a subset Y of a topological space X, we endow it with the subspace topology, also called the relative topology inherited from X. In the first section, we shall see a possible way to define a topology on a set by considering a function from the set to some topological space satisfying specific properties. On the other hand, the same can be done by means of a function from a topological space to our given set. Furthermore, given topological space, not necessarily distinct, we shall see how topologies can be defined on their Cartesian product and their disjoint union (coproduct). In Chapter 2, we saw many topologies on ℝ, such as the usual topology, the lower limit topology, and the K-topology, which were defined by means of a basis. Most of the topologies that we shall see in this chapter are defined by means of a subbasis. It is advisable to refer Section 2.3.2 once before proceeding with this chapter for a better grasp of these concepts.

2.1 Groundwork and Motivation for the Definition

• To begin with, consider the following questions.

  (1) Are the intervals [0, 1] and [0, 5] the same?

  (2) Are the intervals [0, 1] and (0, 1) the same?

  (3) In the plane R2, is a circle the same as a square? Or is a circle of radius 1 the same as the circle of radius 2?

  (4) In R2, is the unit circle the same as the unit disk?

  (5) Is R the same as R2?

  (6) Is the interval (0, 1) the same as R?

  (7) Is the interval (0, 1) the same as the union of intervals (0,#½) ∪ ( #½ , 1)?

  (8) Is the interval [0, 1) the same as the unit circle in R2? Or is (0, 1) or [0, 1] the same as the unit circle in R2?

Of course, to answer this, we must first specify what we mean by the word “same”. Geometrically speaking, clearly, none of the pairs in each of the above questions are the same. For instance, in (1), geometrically the intervals [0, 1] and [0, 5] are of different lengths and as sets [0, 1] ⊆ [0, 5]. However, if we think of the interval [0, 1] as a rubber band of length 1, then we can stretch it from one end to transform it into the interval [0, 5]. Mathematically, we can do this by defining a function f : [0, 1] → [0, 5] by f(x) = 5x. Then we may identify the intervals [0, 1] and [0, 5]. Similarly, in (3), it may not be easy to come up with a function, but if we think of the circle made of a rubber or elastic rope, then we can reshape it into a square to call the circle and a square as “same”. Similarly, we can stretch the circle of radius 1 to form a circle of radius 2. On the other hand, in (7), the interval (0, 1) is a one-piece connected set, but the union (0, ,# ½) ∪ ( #½ , 1) is disconnected as it is missing the point 1 2 . So they do not seem to be the “same” from what we discussed above. In (8), it is difficult to say anything as the interval [0, 1) is a subset of R while the unit circle is in R2.

In the previous chapter, we saw that compactness is a topological property. In this chapter, we shall discuss another important topological property, called connectedness. Vaguely speaking, a connected topological space means a space that is in one single piece. In other words, there is no separation possible for a connected space. The formal definition is given below.

9.1 Connected Spaces

As indicated by the definition, connectedness is a topological property since it is defined in terms of the open sets of X. Therefore, if X is a connected space, then any space that is homeomorphic to X is also connected (see Theorem 9.1.34).

Observe that a separation of a topological space X is a pair of open subsets U and V and U ∩ V = ∅, U ∪ V = X. Thus, it follows that U and V are complements of each other. Since U is open, V = X ∖ U is closed, and since V is open, its complement U = X ∖ V is closed. Alternately, a separation of X can be defined with respect to closed sets as follows.

3.1 Continuous Functions

3.1 Continuous Functions

We are familiar with the notion of continuous function in real analysis, in general, in a metric space. Recall Definition 2.1.7 of continuity of a function in a metric space. We know that a function is continuous on a set if it is continuous at all its points.

Now the question is, how can we define continuity of a function on a set in the absence of a distance function? In Subsection 2.1.2, we discussed that we need to define continuity of a function in terms of open sets in order to define it in a topological space. Let us see how this can be done. We recall the definition in a metric space below and try to see how we can generalize it to a topological space by bringing the open sets into picture.

Let (X, d) and (Y, ρ) be metric spaces and f : X → Y be a function. Then f is continuous if for every a ∈ X and ∈ > 0, there is δ > 0 such that ρ(f(x), f(a)) ∈ whenever d(x, a) δ.

That is,

f(Bd(a, δ)) ⊆ Bρ(f(a), ∈). (3.1)

Equivalently,

Bd(a, δ) ⊆ f−1(Bρ(f(a), ∈ )).

Thus, in order to have continuity of f, we must have Bd(a, δ) ⊆ f−1(Bρ(f(a), ∈ )). In other words, the set f−1(Bρ(f(a), ∈ )) must contain an open ball Bd(a, δ) containing a. That is, a is an interior point of the set f−1(Bρ(f(a), ∈ )). In fact, it can be easily shown that the set f−1(Bρ(f(a), ∈ )) is open in X. Note that the ball Bρ(f(a), ∈ ) is open in Y. Thus, if we have the condition that the “inverse image of every open subset of Y, under f, is open in X”, then we can conclude the continuity of f. This brings the open subsets of X and Y into picture.

We started our discussion in Section 2.1 of Chapter 2 with some motivation from metric spaces to define a topology on a set. Inspired from metric spaces, we defined notions such as open sets and continuous functions in great generality, even in the absence of a metric or a distance function. However, for a metric space, there is a natural way to do this, i.e., we can define a topology on a metric space. The topology so obtained has many desirable properties that are used in analysis. In this chapter, we shall see how a topology is defined on a metric space, and we shall study sequences, continuous functions, etc. on such topological spaces.

5.1 Metric Topology

In this section, we shall see that every metric space is a topological space. Given a metric on a set, we can define topology on it called the metric topology. We begin by giving some examples of metric spaces; their easy verification is left as an exercise as many readers would already be familiar with metric spaces.