Metric and topological spaces pdf

Completions a notcomplete metric space presents the di culty that cauchy sequences may fail to converge. Show that a is closed if and only if it contains all its limit points. If x is a topological space and x 2 x, show that there is a connected subspace k x of x so that if s is any other connected subspace containing x then s k x. Explain what is m eant by the interior int a and the closure a of a. Introduction in chapter i we looked at properties of sets, and in chapter ii we added some additional structure to a set a distance function to create a pseudomet. That is, in topological spaces, instead of deriving the notion of open and closed sets from a predefined metric, we. What topological spaces can do that metric spaces cannot. There are many ways to make new metric spaces from old. Introduction when we consider properties of a reasonable function, probably the. N and it is the largest possible topology on is called a discrete topological space. Click download or read online button to get introduction to metric and topological spaces book now. A topological space whose topology can be described by a metric is called metrizable. We then looked at some of the most basic definitions and properties of pseudometric spaces.

W a sutherland, introduction to metric and topological spaces, clarendon press metric and topological spaces course hero introduction to. Possibly a better title might be a second introduction to metric and topological spaces. Recall from singlevariable calculus that a function f. 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. This is an ongoing solutions manual for introduction to metric and topological spaces by wilson sutherland 1. Such a family o is referred to as a topology for x. The main reason for taking up such a project is to have an electronic backup of my own handwritten solutions. A topological space whose topology is a metric topology is called a metrizable space. Semantic scholar extracted view of introduction to metric and topological spaces by wm. 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. If x,d is a metric space we call the collection of open sets the topology induced by the metric. Knebusch and their strictly continuous mappings begins. 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.

A metric space m is compact if every sequence in m has a subsequence that converges to a point in m. This new edition of wilson sutherlands classic text introduces metric and topological spaces by describing some of that influence. Most of the funda mental examples of topological spaces. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. Distance functions lead to metric spaces, which is how we usually think of spaces. Leif mejlbro topological and metric spaces, banach spaces and. 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. A limit point of a is a point x 2 x such that any open neighbourhood u of x intersects a. It is not hard to check that d is a metric on x, usually referred to as the discrete metric. Examples of topological spaces john terilla fall 2014 contents 1 introduction 1 2 some simple topologies 2 3 metric spaces 2 4 a few other topologies on r1 and r2. Namely, we will discuss metric spaces, open sets, and closed sets. 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. Introduction to metric and topological spaces paperback. Introduction to metric and topological spaces download.

Reasonably, we want to repair this situation, and in as economical way as possible. 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. Please note, the full solutions are only available to lecturers. In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. It is assumed that measure theory and metric spaces are already known to the reader. The aim is to move gradually from familiar real analysis to abstract topological spaces, using metric spaces as a bridge between the two. Thenfis continuous if and only if the following condition is met.

First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological 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. Introduction to topological spaces and setvalued maps. 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 8 connected spaces 46 9 product spaces 51 10 urysohns and tietzes theorems 57 11 appendix 60 3. Details of where to hand in, how the work will be assessed, etc.

What is the difference between topological and metric spaces. In this paper a systematic study of the category gts of generalized topological spaces in the sense of h. Xthe number dx,y gives us the distance between them. This site is like a library, use search box in the widget to get ebook that you want. We then have the following fundamental theorem characterizing compact metric spaces. Metricandtopologicalspaces university of cambridge. Introduction to metric and topological spaces by wilson. Neighbourhoods, closed sets, convergence and continuity.

Let be a mapping from to we say that is a limit of at, if 0 0, there exists q2 s\ bp. Introduction by itself, a set doesnt have any structure. The language of metric and topological spaces is established with continuity as the motivating concept. Suppose fis a function whose domain is xand whose range is contained in y. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. If x,d is a metric space and a is a nonempty subset of x, we can make a metric d a on a by putting.

Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Two metric spaces that are related by an isometry are equivalent from the point of view of metric space theory. A metric induces a topology on a set, but not all topologies can be generated by a metric. Clark we assume that the reader has a good working familiarity with the notion of a metric space, but to. Department of pure mathematics and mathematical statistics, university of cambridge. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to help convey familiarity.

This is known as sequential compactness and, in metric spaces but not in general topological spaces, is equivalent to the topological notions of countable compactness and compactness defined via open covers. Sutherland metric and topological spaces part 1 university of york mathsoc the aim of the module is to introduce students to metric spaces to show how. The first goal of this course is then to define metric spaces and continuous. General topology 1 metric and topological spaces the deadline for handing this work in is 1pm on monday 29 september 2014. While in topological spaces the notion of a neighborhood is just an abstract concept which reflects somehow the properties a.

The discussion above ensures what computer scientists call downward compatibility. Introduction to metric and topological spaces oxford. All topologies come from a family of 01 valued quasimetrics. 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. 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. Department of pure mathematics and mathematical statistics. We do not develop their theory in detail, and we leave the veri. A metric space x, d is a space x with a distance function d. Solomon lefschetz in order to forge a language of continuity, we begin with familiar examples.