I think the OP is confusing a vector space with a normed vector space,which indeed shares many properties of general metric spaces.And for a very simple reason: A norm induces a metric on the underlying set on which the map is defined.This is fairly simple to prove from the definitions and the questioner should try and do it.

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.

A good book for metric spaces specifically would be Ó Searcóid's Metric Spaces. However, note that while metric spaces play an important role in real analysis, the study of metric spaces is by no means the same thing as real analysis. A good book for real analysis would be Kolmogorov and Fomin's Introductory Real Analysis.

2017年7月26日 · I understand that the above definition of continuity in metric spaces is a very intuitive extension of the $\epsilon-\delta$ version. However, what I don't understand is that the above definition depends on what metric we are using in each metric space right?

The second definition stated that a metric space $(D, \rho)$ is discrete if there exists some $\kappa > 0 ...

The best I can think of, are: Given a metric space $(X,d)$, we can assign sigma-algebras. Borel Measure: This is the sigma algebra generated by the open sets generated by the open balls in the metric. Or, 2) Looking at some of the linked questions are that of a Hausdorff measure associated with a metric space:

This is to ensure the triangle inequality. In your proposal, it can happen that $[p]$ and $[q]$ have nearby representatives and $[q]$ and $[r]$ have nearby representatives, but the two representatives of $[q]$ involved are different, so this doesn't guarantee that $[p]$ and $[r]$ have nearby representatives, so the triangle inequality may be violated.

$\begingroup$ It is not enough to show that you can associate a topological space with a metric space to show that a metric space is a topological space. After all, I can also associate a pink elephant with a metric space. That doesn't show that a …

For one implication you have a countable base of your space. Take a point in every element of the base. Since every open set is union of elements of the base, every open sets contains one of the choosen points...

2015年5月29日 · The metric structure in a normed space is very special and possesses many properties that general metric spaces do not necessarily have. Metric spaces are also a kind of a bridge between real analysis and general topology. With every metric space there is associated a topology that precisely captures the notion of continuity for the given metric.