Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

closed as off-topic by Jonas Meyer, Alan, Mark Fantini, Adam Hughes, John Mar 24 at 3:59

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jonas Meyer, Alan, Mark Fantini, Adam Hughes, John
If this question can be reworded to fit the rules in the help center, please edit the question.

Take the indiscrete topology on a non-trivial space. –  Alex Youcis Feb 19 '12 at 20:55
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.

share|improve this answer

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 $\varnothing$.

share|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.