Uniform continuity in top. spaces

1. May 8, 2010

Bleys

So my teacher said that uniform continuity was a metric space notion, not a topological space one. At first it seemed obvious, since there is no "distance" function in general topological spaces. But then I remembered that you can generalize point-wise continuity in general topologies, so why not uniform continuity? I'm probably wrong; I wouldn't know the details to construct such a definition, but I know that Hausdorff spaces behave similarly to metric spaces (being Hausdorff themselves) in some respects, so maybe it would be possible?

2. May 8, 2010

eok20

The definition of a continuous function from a topological space X to a topological space Y is that the inverse image of an open set in Y is an open set in X. This agrees with the epsilon delta definition of continuity for a metric space.

In a metric space, the idea of uniform continuity is that for a given epsilon, for any point in the domain you can pick the same size delta ball and the function will map into an epsilon ball around the image of that point. So if you wanted to extend this idea to a general topological space, you would need some notion of open sets around different points being of a comparable size. To do this you need some additional structure, and this is the idea of a uniform space: http://en.wikipedia.org/wiki/Uniform_space

3. May 9, 2010

g_edgar

There is a more general type of space, called uniform space ... As with the metric case, a single toplogical space may correspond to many different uniform spaces.

4. May 9, 2010

Bacle

Just to add the side statement that if the space is metrizable, then the
metric coincides with its uniformity.

5. May 11, 2010

g_edgar

To contradict Bacle, two different metrics may yield the same uniformity.
For example, in the real line, take distance |x-y| in one case, and distance 2|x-y| in the other case.

6. May 17, 2010

Bacle

Right, my bad. Up to equivalent metrics.