MA651 Topology. Lecture 3. Topological spaces.This text is based on the following books:• ”Linear Algebra and Analysis” by Marc Zamansky• ”Topology” by James Dugundgji• ”Elements of Mathematics: General Topology” by Nicolas BourbakiI have intentionally made several mistakes in this text. The first homework assignment is to findthem.A topological space has a structure in which the concepts of limit and continuous function can bespecified.Though R1and Rnare obviously different, this is not due to one having more points thanthe other, since we know that card R1= card Rn. Geometrically, it is evident that the points arearranged differently, so that different subsets are ”close together”. To detect inherent differencesof this sort, we study sets in which a notion of ”nearness” is specified, that is, in which topologyis specified.All the fundamental concepts of topology derive from properties of real numbers and real-valuedfunctions of a real variable. However, the real numbers have a very rich structure; their propertiesstem from many fundamental notions: total order relations, group, ring and field properties, theconcept of an absolute value, the existence of the rational numbe rs forming a dense set, the factthat every Cauchy sequence converges, etc.If we consider, for example, continuous functions, we see that some properties are true when thevariable lies in an interval (open or not), and that others are true only if the interval is closed.We have therefore tried to prove our theorems under the most general hypotheses, and are led toformulate the basic concepts required for their proof.The concept of a continuous function enables us to give a sense to the expression ”f(x) tendsto y0when x tends to x0”. This concept of a limit proves inadequate. Further, the concept ofa countable convergent sequence which is the appropriate concept in the study of metric spacesproves inadequate for non-metric spaces. Finally, we find it desirable to have a unified way ofdescribing such diverse phenomena as the following: f(x) tends to y0when x tends to x0, f (x)tends to y0when x tends to x0on the right (or on the left), xntends to y0when n tends to infinity,1situations which are not always related to a concept of continuity.We arrive at axiomatic definitions of open sets, neighborhoods, filter bases, etc. This method,although perhaps unfamiliar, has the advantage of enabling us to derive certain properties fromthe minimum hypotheses. These concepts have their origins, directly or indirectly, in familiarnotions taken from the real line. The usual way of developing of the subject consists in definingopen sets, then neighborhoods of a point, or firstly neighborhoods and then open sets; the generalconcept of a limit is introduced afterwards, if needed.Now the principal concepts taken from properties of the real line depend on the fact that weuse open intervals (we can even consider only intervals whose points are rational numbers), andif we consider the set of open intervals it is clear that, if X and Y are two such intervals, theirintersection XTY contains another open interval (the empty set being also considered as an openinterval).This observation is even more illuminating in the case of the plane. When we study, for example,the concept of continuity or limit in the plane we need only make use of open discs. Now theintersection of two open discs is either empty, or is a set which, although not itself an open disc,contains one. In our development of the subject this property, elevated to the status of an axiom,plays a fundamental role.19 ”Local” Definition of TopologyIn this section we define topology starting from elements of a set E, while in the next section wewill define topology as a collection of subsets of E, in fact, both definitions are equivalent.19.1 Fundamental FamiliesThe fundamental idea consist in the fact that in questions of topology or of limits we have to dowith a family Ω of subsets X of set E such that the intersection X ∩ X0of two elements of Ωcontains another element X00of Ω. However if we want to use this observation as an axiom wemust note that the empty set Ø ∈ Ω, since for every subset A of E we have Ø ⊂ A, the conditionthat X ∩ X0contains X00will always be satisfied by taking X00= Ø. We shall have to make ourcondition more precise.On the other hand the empty set will play an important part. In fact, in broad terms, the propertyØ 6∈ Ω is related to the concept of limit, and the property Ø ∈ Ω is related to that of topology.We agree, once and for all, that we shall not consider empty families.Definition 19.1. A fundamental family on a set E is a non-empty family Ω of subs ets of E suchthat if X and X0belong to Ω, X ∩ X0contains an element X00of Ω, and X006= Ø if X ∩ X06= Ø.This definition gives rise to the f ollowing observations:21. If Ω is a fundamental family on a set E, if it contains non-empty sets, and Ø ∈ Ω, it neednot contain non-empty sets X, X0such that X ∩ X0= Ø. For example, we may take Ω toconsist of Ø and E.2. If a non-empty family Ω is a fundamental family, and if Ø 6∈ Ω, then no element of Ωis empty, and two (and so any finite number of) arbitrary elements of Ω have non-emptyintersection.3. If Ω is a given family not containing Ø, to prove that Ω is a fundamental family it sufficesto show that for two arbitrary elements X, X0, X ∩ X0contains an X00∈ Ω.4. If Ω is a family containing two non-empty elements X, X0such that X ∩ X0= Ø, in orderthat Ω shall be a fundamental family we must have Ø ∈ Ω.5. The more restrictive condition X ∩ X0∈ Ω also defines a fundamental family.6. A family consisting of a single subset of a set is a fundamental family, but is without interest.7. If Ω is a fundamental family which does not contain Ø, we again have a fundamental familyon adjoining Ø to Ω.19.2 Properties of Fundamental Families1. If Ω is a fundamental family, every finite intersection of its members contains an element ofΩ, and this element is not empty if the intersection is not empty.2. Let Ω be a fundamental family on a set E, and f a mapping of E into a set F . Let Y bethe family of subsets of F consisting of the f(X) where X ∈ Ω.(a) In general Y is not a fundamental family. We give an example of this.Let A, A0be disjoint intervals of R, and B, B0two distinct non-disjoint intervals of R.Using restrictions of linear