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I am taking a course in intro topology and very confused because the lecturer has only given us a definition of metric topology while all the texts I have read have the axiomatic definition of topology.

I don't think they are the samee but what is the difference? Also is metric topology same as 'discrete topology'?

Definitions I have are:

Metric topology: Collection of open sets such that all subsets are in X.

Topology: The three axioms of it should contain X, empty set, union and intersectoin should be in the topology.

Discrete topology: T is the collection of all subsets of a non empty set X.

To me discrete and metric look the same?

I am not understanding anything in lectures so I am trying to teach myself with resources online. Thanks for your help.

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    $\begingroup$ Simple answer is 'no', they are different. A metric topology is defined simply as a topology with a metric defined on it. Not all topologies have metrics. Not all metric spaces are topologies either, necessarily. The definitions of these things are different. If your lecturer is defining topologies in terms of metrics or metrics in terms of topologies then he shouldnt be teaching the subject. $\endgroup$
    – SquishyRhode
    Commented Feb 8, 2020 at 16:27
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    $\begingroup$ A metric topology is a special kind of topology. $\endgroup$
    – Thorgott
    Commented Feb 8, 2020 at 16:28
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    $\begingroup$ Please be a bit more specific. When you say "the axiomatic definition of topology", are you referring to this definition of a topological space? I have no clue what you mean by "a definition of metric topology". $\endgroup$
    – Ben Grossmann
    Commented Feb 8, 2020 at 16:29
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    $\begingroup$ @SquishyRhode that seems like an overly bold statement. Talking about metric spaces is a fine way to begin a course on topology. $\endgroup$
    – Ben Grossmann
    Commented Feb 8, 2020 at 16:31
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    $\begingroup$ My point is that they are definitionally distinct ideas. Any overlap or equivalence is a proven thing rather than a definitional one. $\endgroup$
    – SquishyRhode
    Commented Feb 8, 2020 at 16:33

2 Answers 2

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A metric $d$ on $X$ lets us define a notion of open sets, i.e. a topology $\mathcal{T}_d$. The latter obeys the abstract axioms of a topology.

But many abstract topologies cannot have a metric defined on them that defines that topology in this way. There are theorems on when this is and is not the case. Many topologies in practical maths arise from metrics, so in the beginning there is often more focus on metrics (analysis has a lot of use for it). E.g. at my university we had a course in "metric topology" in year 1, introductory general topology in year 2, advanced topics later.

The discrete topology is actually a general topology that is generated by a metric: $d(x,y)=1$ when $x \neq y$, $d(x,x)=0$. The usual topology on $\Bbb R$, $\Bbb Q$ etc. all arise from metrics.

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A topology $\tau$ on a set $X$ is a collection of subsets of $X$ satisfying three conditions (and you know what conditions they are).

In a metric space $(X,d)$, we can define open balls $B_r(x)$ to be $$B_r(x)=\{y\in X: d(x,y)<r\}.$$

Now the collection of all subsets of $X$ which are called open balls, $\{B_r(x)\}$, satisfy the three conditions of being a topology (in fact a basis for a topology) and this topology is called "metric topology" or "the topology induced from a metric". Note that not all topologies induced from a metric, e.g the trivial topology $\tau=\{\varnothing, X\}$.

We also have phrases like "partition topology" which a partition on a set is given and the subsets of the partition satisfy the three conditions.

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  • $\begingroup$ Does that mean the definitions mean the same? Except the last part that not all topologies can be induced by metrics. Also how is the trivial topology not induced by a metric? $\endgroup$
    – NewYork
    Commented Feb 8, 2020 at 16:37
  • $\begingroup$ Also what about discrete topology? The definiton is very similar. $\endgroup$
    – NewYork
    Commented Feb 8, 2020 at 16:40
  • $\begingroup$ @NewYork the special thing about the discrete topology is that $\tau$ contains every subset of the space $X$. Usually, $\tau$ contains only certain subsets, which is to say that no every subset of $X$ is open. $\endgroup$
    – Ben Grossmann
    Commented Feb 8, 2020 at 17:11
  • $\begingroup$ The trivial topology is metrizable if $X$ has a single point (or no points). $\endgroup$
    – user76284
    Commented Feb 9, 2020 at 1:01

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