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Give an example of a topological space which is not a metric space, i.e. whose topology is not associated with any metric.

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Take the indiscrete topology on a non-trivial space. –  Alex Youcis Feb 19 '12 at 20:55
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Is this homework? Anyway, think of the simplest possible topology. What is it? Does it come from a metric space? –  abatkai Feb 19 '12 at 20:56

2 Answers 2

A topological space whose topology can be associated with a metric is called metrizable. There are many necessary conditions for being metrizable; an easy one is that a space must be Hausdorff if it is metrizable, so any non-Hausdorff space cannot have a topology that comes from a metric.

The set $\{0,1\}$ with the trivial topology is a simple example of a non-Hausdorff space.

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Metric topologies has many properties which are not needed from a general topological space:

  1. Every point $x$ is such that $\{x\}$ is closed;
  2. For every $x,y$ there are $U,V$ open and disjoint such that $x\in U$ and $y\in V$;
  3. If $A$ is a set, then $x$ is in the closure of $A$ if and only if there exists a sequence in $A$ which converges to $x$.

And many many more.

Whereas a topology need only to satisfy three axioms:

  1. The union of open sets is open;
  2. The intersection of finitely many open sets is open;
  3. The space and the empty set are open.

If we take a set $A$ then it has many topologies which are not metric, for example if we ensure that $\{x\}$ is not closed, or if we cannot separate points, and so on.

An example is $A$ which has more than two points with the topology in which the only open sets are $A$ and $\varnothing$.

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