A metric space is a set xtogether with a metric don it, and we will use the notation x. If a subset of a metric space is not closed, this subset can not be sequentially compact. Xcan also be an arbitrary nonempty subset of c, for example x r. The equivalence of these three concepts is not true in a general topological space. Then we call k k a norm and say that v,k k is a normed vector space. If x,d is a metric space and y is a nonempty subset of x, then dy x,y dx,y for all x,y. A topological space is an aspace if the set u is closed under arbitrary intersections. Xthe number dx,y gives us the distance between them. A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. The nagatasmirnov metrization theorem extends this to the nonseparable case. For instance, consider the real numbers with an infinitesimal positive element. In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. It turns out that a great deal of what can be proven for.
Metric and topological spaces a metric space is a set on which we can measure distances. Xx from the case when x is a compact metric space to the case when x is allowed to be noncompact. Topological space, euclidean space, and metric space. What is the difference between topological and metric spaces. A subset with the inherited metric is called a submetric space or metric subspace. It states that a topological space is metrizable if and only if it is regular, hausdorff and has a. Topological entropy for nonuniformly continuous maps boris hasselblatt, zbigniew nitecki, and james propp abstract. X can be joined by a continuous path of length dx,y. Introduction to topological spaces and setvalued maps. In his discussion of metric spaces, we begin with euclidian nspace metrics, and move on to discrete metric. A topological space is called separable if it has a countable dense subset. A particular case of the previous result, the case r 0, is that in every metric space singleton sets are closed. Metric spaces, topological spaces, products, sequential continuity and nets, compactness, tychonoffs theorem and the separation axioms, connectedness and local compactness, paths, homotopy and the fundamental group, retractions and homotopy equivalence, van kampens theorem, normal. If v,k k is a normed vector space, then the condition du,v ku.
Deduce that every subspace of a separable metric space is separable. U nofthem, the cartesian product of u with itself n times. Often, if the metric dis clear from context, we will simply denote the metric space x. A particular case of the previous result, the case r 0, is that in. Indeed let x be a metric space with distance function d. A subset is called net if a metric space is called totally bounded if finite. Any topological space can be converted into a metric space only if there is a. 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.
Free topology books download ebooks online textbooks tutorials. So, consider a pair of points one meter apart with a line connecting them. The topological properties of metric spaces can be expressed entirely in terms. N and it is the largest possible topology on is called a discrete topological space. In fact, a metric is the generalization of the euclidean metric arising from the four longknown properties of the euclidean distance. A totally bounded metric space is bounded, but the converse need not hold. Topologytopological spaces wikibooks, open books for an. The topologies are discrete and indiscretethere is no gossipy topology. In a metric space, you have a pair of points one meter apart with a line connecting them. While topology has mainly been developed for infinite spaces, finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding.
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. Sep 24, 2015 metric spaces have the concept of distance. Several concepts are introduced, first in metric spaces and then repeated for. In mathematics, a finite topological space is a topological space for which the underlying point set is finite. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Definitions and examples 5 d ax,y dx,y for all x,y. That is, a topological space, is said to be metrizable if there is a metric. The language of metric and topological spaces is established with continuity as the motivating concept. However, work in cognitive psychology has challenged such simple notions of sim ilarity as models of human judgment, while applications frequently employ non euclidean distances to measure object similarity. Any normed vector space can be made into a metric space in a natural way. Another interest in nonmetrizable spaces comes from the theory of quasi metrics 16 and. Namely, we will discuss metric spaces, open sets, and closed sets. 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. That is, it is a topological space for which there are only finitely many points.
It is assumed that measure theory and metric spaces are already known to the reader. Topology of metric spaces gives a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas to encourage geometric thinking, to treat this as a preparatory ground for a general topology course, to use this course as a surrogate for real analysis and to help the students gain some. Introduction to metric and topological spaces oxford. If v,k k is a normed vector space, then the condition. A metric space gives rise to a topological space on the same set generated by the open balls in the metric. X, then y with the same metric is a metric space also. Paper 2, section i 4e metric and topological spaces. A subset is called net if a metric space is called totally bounded if finite net. Since every metric space is hausdorff, every compact subset is also closed. The empty set and x itself belong to any arbitrary finite or infinite union of members of.
Product topology the aim of this handout is to address two points. 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. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Metric spaces, topological spaces, products, sequential continuity and nets, compactness, tychonoffs theorem and the separation axioms, connectedness and local compactness, paths, homotopy and the fundamental group, retractions and homotopy equivalence, van kampens theorem, normal subgroups, generators and. Introduction when we consider properties of a reasonable function, probably the.
The source of confusion in the definitions usually has to do with the definition of an open set. Jul 15, 2010 we shouldnt ask the difference between a metric space and a topological space, indeed its been mentioned above that sometimes they are the same, and that every metric space is a topological space. Metricandtopologicalspaces university of cambridge. If xis compact as a metric space, then xis complete as we saw in lecture and totally bounded obvious. This forms a topological space from a metric space. What topological spaces can do that metric spaces cannot82 12. Corollary 9 compactness is a topological invariant. Classification in nonmetric spaces 839 to considerable mathematical and computational simplification. Closed sets, hausdorff spaces, and closure of a set. The most familiar metric space is 3dimensional euclidean space.
These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. There are many ways to make new metric spaces from old. Chapter 9 the topology of metric spaces uci mathematics. We shouldnt ask the difference between a metric space and a topological space, indeed its been mentioned above that sometimes they are the same, and that every metric space is a topological space. If x,d is a metric space and a is a non empty subset of x, we can make a metric d a on a by putting. R r is an endomorphism of r top and of r san, but not.
It is not hard to check that d is a metric on x, usually referred to as the discrete metric. What is the difference between metric space and topological. A topological space is separable and metrizable if and only if it is regular, hausdorff and secondcountable. Free topology books download ebooks online textbooks. Metric space versus topological space physics forums. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. If for a topological space, we can find a metric, such that, then the topological space is called metrizable. The space of tempered distributions is not metric although, being a silva space, i. Any metric space may be regarded as a topological space.