MetricSpaces2.wmv

Metric Spaces Part 2 Concepts and Tools: | Limit Points | Sequences of Points | Convergence | The Cauchy Criterion | Cauchy Sequences | Complete Metric Spaces | Incomplete Metric Spaces | Rational Numbers as an Incomplete Metric Space | The Cardinality of Incomplete and Complete Metric Spaces | Open Sets | Interior Points | Construction of Topologies |

[magicguy]

In actuality, you would say that the infinite series for Euler's number in the metric space (Z, d_z) is divergent because its limit is not in Z, its sequence of partial sums is a Cauchy sequence in Z. A much easier to understand example of an incomplete metric space is: let X= (0, 1) and d_x be arbitrary metric. It is incomplete bc a_k=1/k is Cauchy but has no limit on X

[Mathview]

@66mrudula In the case of differential manifolds we have the Riemann metric which is symmetric and positive definite. Hence, all non-zero vectors have positive length. It's also possible to define a semi-Riemann metric or pseudo-Riemann metric (like that of special relativity) that allows some vectors to have negative length, or any real number as length. An example of a semi-Riemann metric would be g = diag(1, -1, -1, -1) with space-time coordinates (t, x, y, z).

[Mathview]

@LadislauFernandes Ah.. Thank you. I followed your suggestion, and just now posted a YouTube Playlist called: Metric Spaces. Just go to the Mathview Channel and click on Playlists at the top. Look for the title Metric Spaces. There are six or seven relevant videos in more or less logical order.

[LadislauFernandes]

1. Very nice job you are doing here. You are making a huge difference. Congrats and many many tks
2. Are these metricSpaces videos in a list? if not, please do a list.
3. How come the video lasts for 16.45 minutes?