General topology 1 metric and topological spaces the deadline for handing this work in is 1pm on monday 29 september 2014. Every function from a discrete metric space is continuous at every point. That is, a topological space, is said to be metrizable if there is a metric. Lets then define what open subsets of a metric space are. Any topological space can be converted into a metric space only if there is a. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Introduction to metric and topological spaces oxford mathematics pdf one of the ways in which topology has influenced other branches of mathematics in the past few decades is by putting the study of continuity and convergence into a general setting.
Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Several concepts are introduced, first in metric spaces and then repeated for. In this section we briefly overview some basic notions about metric spaces and topology. Metric and topological spaces contents 1 introduction 4 2 metric spaces 5 3 continuity 17 4 complete spaces 22 5 compact metric spaces 35 6 topological spaces 40 7 compact topological spaces 44. Then we say that d is a metric on x and that x, d is a metric space. Metric and topological spaces part ib of the mathematical tripos of cambridge this course, consisting of 12 hours of lectures, was given by prof. First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological spaces. Ifa is a metric space, we replace it by the associated topological space.
Metric spaces, topological spaces, and compactness 253 given s. N and it is the largest possible topology on is called a discrete topological space. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. This chapter will introduce the reader to the concept of metrics a class of functions which is regarded as generalization of the notion of distance and metric spaces.
It is assumed that measure theory and metric spaces are already known to the reader. Further it covers metric spaces, continuity and open sets for metric spaces, closed sets for metric spaces, topological spaces, interior and closure, more on topological structures, hausdorff spaces and compactness. In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. In a metric space, you have a pair of points one meter apart with a line connecting them. The level of abstraction moves up and down through the book, where we start with some realnumber property and think of how to generalize it to metric spaces and sometimes further to general topological spaces. Clark we assume that the reader has a good working familiarity with the notion of a metric space, but to. The idea of a topological space is to just keep the notion of open sets and abandon metric spaces, and this turns out to be a really good idea. That doesnt show that a metric space is a pink elephant. A bounded closed subset of is sequentially compact, by heineborel theorem. Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about topological spaces also apply to all metric spaces. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. Any normed vector space can be made into a metric space in a natural way. A topological space whose topology is a metric topology is called a metrizable space. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur.
We also have the following simple lemma lemma 3 a subset u of a metric space is open if and only if it is a neighbor hood of each of its points. Click download or read online button to get introduction to metric and topological spaces book now. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. This site is like a library, use search box in the widget to get ebook that you want. This is an ongoing solutions manual for introduction to metric and. A metric space is a set x together with a distance function d. A metric space is called sequentially compact if every sequence of elements of has a limit point in.
An open sets family of a metric space is defined next and it has been shown that the metric space with any open sets family is a topological space. So, consider a pair of points one meter apart with a line connecting them. There are many papers on the subject, but few of any. Introduction to topological spaces and setvalued maps. Details of where to hand in, how the work will be assessed, etc. Sutherland, introduction to metric and topological spaces particularly a topological space x, t consists of a nonempty set x together with a family t. Indeed let x be a metric space with distance function d. All topologies come from a family of 01 valued quasimetrics. Introduction to topology answers to the test questions stefan kohl question 1. After all, i can also associate a pink elephant with a metric space. Semantic scholar extracted view of introduction to metric and topological spaces by wm. Kevin mahardika introduction to metric and topological spaces pdf. Introduction to metric and topological spaces download.
Metricandtopologicalspaces university of cambridge. A topological space is the most basic concept of a set endowed with a notion of neighborhood. Any metric space can be converted into a topological space such that an open ball in a metric space corresponds to a basis in the corresponding topology metric spaces as a specialization of topological spaces. The book assumes some familiarity with the topological properties of the real line, in particular convergence and completeness.
What topological spaces can do that metric spaces cannot. A metric space x, d is a space x with a distance function d. Most of the funda mental examples of topological spaces. Introduction when we consider properties of a reasonable function, probably the. A subset u of a metric space x is closed if the complement x \u is open. Introduction to metric and topological spaces oxford. Possibly a better title might be a second introduction to metric and topological spaces. In contrast, we will also discuss how adding a distance function and thereby turning a topological space into a metric space introduces additional concepts missing in topological spaces, like for example completeness or boundedness. Every metric space can also be seen as a topological space. A topological space x is a pair consisting of a set xand a collection. We do not develop their theory in detail, and we leave the veri. Any metric space may be regarded as a topological space.
In this section we will be studying the concept of neighborhood or closeness in generic metric. We recall that a subset v of x is an open set if and only if, given any point vof v, there exists some 0 such that fx2x. Despite sutherlands use of introduction in the title, i suggest that any reader considering independent study might defer tackling introduction to metric and topological spaces until after completing a more basic text. The language of metric and topological spaces is established with continuity as the motivating concept. Metric spaces, topological spaces, and compactness proposition a. A topological space is an aspace if the set u is closed under. On few occasions, i have also shown that if we want to extend the result from metric spaces to topological spaces, what kind of extra conditions need to be imposed on the topological. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3.