Content copied here again

Let (E, d) be a metric space, then

How to prove that T is a topology? tbd

So from metric space, topology can be derived, but not necessarily other way around. If so, it is called metrizable topology. For example, trivial topology which contains only empty set and the full set is not metrizable.

So in Hausdorif topological space, we

For example, Xn = 1/n², how to prove it converges to 0?

The framework is to let epislon < 0, leading to d(Xn, 0) < epislon. Hence the following

Next, the notion of Cauchy sequence, the definition is

If all Cauchy sequence of elements of X converge, then this metric space X is called complete. For example, R (real number space) is complete, but Q (quotient) is not. As a fact, R is defined as a way to complete Q!

Following the thinking logic to prove Xn = 1/n^2 converges to 0, we can easily prove it is also a Cauchy sequence.