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What is the difference between a topological and a metric space?

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Don't yell! (Capital letters are interpreted as shouts). And please make your titles informative. "I want to learn" is not informative. – Arturo Magidin Feb 10 '11 at 5:19
Metric spaces are a specific type of topological space, with many nice properties. They are much easier to understand intuitively, and general enough for many applications. – Yuval Filmus Feb 10 '11 at 5:20
Every metric space is a topological space, but not every topological space is a metric space. Do you know the definitions, or are you just now encountering them? – Arturo Magidin Feb 10 '11 at 5:21
@Arturo: A nitpick. I have heard this said by many people "Every metric space is a topological space". I would actually prefer to say every metric space induces a topological space on the same underlying set. – user17762 Feb 10 '11 at 6:30
@Sivaram: Fair enough. I say it in much the same sense that I say "Every ring is an abelian group", when I should "really" say "there is a faithful forgetful functor from Rings to AbGroups which commutes with the forgetful functors to Set"... (-: – Arturo Magidin Feb 10 '11 at 6:33

Just in terms of ideas: a metric space has a notion of distance, while a topological space only has a notion of closeness. If we have a notion of distance then we can say when things are close to each other. However, distance is not necessary to determine when things are close to each other.

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A topological space is a set $X$ along with another set usually denoted by $\tau$ which is a collection of subsets of $X$, satisfying the following properties:

  1. $\emptyset, X \in \tau$
  2. Countable or Uncountable union of sets in $\tau$ is again in $\tau$
  3. Finite intersection of sets in $\tau$ is again in $\tau$

The space $(X,\tau)$ is called the topological space and the set $\tau$ is called a topology on $X$. The elements of $\tau$ are called open sets.

A metric space is a set $X$ and a function $d:X \times X \rightarrow \mathbb{R}^+ \cup \{0\}$ called the "metric" which takes in two elements from the set and pops out a non-negative real number. This metric has to satisfy certain properties:

  1. $d(x,y) \geq 0$, $\forall x,y \in X$
  2. $d(x,y) = 0$, iff $x=y$
  3. $d(x,y) = d(y,x)$, $\forall x,y \in X$
  4. $d(x,y) \leq d(x,z) + d(z,y)$, $\forall x,y,z \in X$

The space $(X,d)$ is called the metric space and $d$ is the metric i.e. a function such that $d:X \times X \rightarrow \mathbb{R}^+ \cup \{0\}$

Using this metric, we can define "certain" sets. The set of these sets call it $\tau$, along with the original set $X$ can now shown to be a topological space. So with every metric space $(X,d)$, we can associate a topological space $(X,\tau)$. The elements of the set $\tau$ are open sets.

However, topological spaces need not arise out of a metric space. There are non-metrizable topological spaces.

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@Sivaram: I think there is a theorem which asserts that when a topological space is metrizable. If there is such a theorem, then you may consider adding that to the an answer as well. – anonymous Feb 10 '11 at 5:54
@Chandru1: There are some metrizable theorems. But I do not know them well enough to write them out here. – user17762 Feb 10 '11 at 5:58
You could also add an example of a non-metrizable topology, to wit, the discrete topology. – Jose L. Lykón Feb 10 '11 at 5:59
@Jose: The discrete topology arises from the discrete metric: $d(x,y) = 0$ if $x=y$, $d(x,y)=1$ if $x\neq y$. So the discrete topology is certainly metrizable. (The indiscrete topology arises from the pseudo-metric $d(x,y)=0$ for all $x,y$; no metric defines the indiscrete topology on a space with more than one point, but pseudo-metrics are usually "good enough"). – Arturo Magidin Feb 10 '11 at 6:03
@Jose: As Arturo points out, discrete topology arises out of a discrete metric. – user17762 Feb 10 '11 at 6:28

A metric space gives rise to a topological space on the same set (generated by the open balls in the metric). Different metrics can give the same topology. A topology that arises in this way is a metrizable topology. Using the topology we can define notions that are purely topological, like convergence, compactness, continuity, connectedness, dimension etc. Using the metric we can talk about other things that are more specific to metric spaces, like uniform continuity, uniform convergence and stuff like Hausdorff dimension, completeness etc, and other notions that do depend on the metric we choose. Different metrics that yield the same topology on a set can induce different notions of Cauchy sequences, e.g., so that the space is complete in one metric, but not in the other. In analysis e.g. one often is interested in both of these types of notions, while in topology only the purely topological notions are studied. In topology we can in fact characterize those topologies that are induced from metrics. Such topologies are quite special in the realm of all topological spaces. So in short: all metric spaces are also topological spaces, but certainly not vice versa.

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An important difference in terms of technique is that in a metric space there are distinguished neighborhoods, namely the open balls of radius $r$ around a point $x$: $$ B_r(x) = \{y\ :\ d(x,y)<r\}. $$ These open balls form a local base for the topology and hence, carry all information about the topology of the metric space.

While in topological spaces the notion of a neighborhood is just an abstract concept which reflects somehow the properties a "neighborhood" should have, a metric space really have some notion of "nearness" and hence, the term neighborhood somehow reflects the intuition a bit more.

Moreover, in a metric space it is more convenient to work with sequences than in a topological space. For example it makes total sense, to memorize convergence of a sequence $(x_n)$ in a metric space to a point $x$ as "from some point on all $x_n$ are arbitrarily close to $x$". A statment which is quite useless in a topological space.

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If metric space is interpreted generally enough, then there is no difference between topology and metric spaces theory (with continuous mappings). Building on ideas of Kopperman, Flagg proved in this article that with a suitable axiomatization, that of value quantales, every topological space is metrizable. This gives rise to a precise equivalence between the category of topological spaces and the category of generalized metric spaces, presented here (alg. univ.).

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