4

The diagram below is meant to represent some type of relationship between metric spaces of various types, or subspaces of said metric spaces of various types. The diagram itself is still a very early draft, so as time goes on, I will likely add more and more details to the diagram, such as whether or not various relationships are applicable only to subspaces, or to the entire metric space in question. At the moment, I'm simply for feedback regarding whether or not relationships the likes of which the diagram in it current state suggests actually exist and are correct.

enter image description here

Any feedback is appreciated, thank you.

joshuaheckroodt
  • 2,789
  • 2
    A complete subspace of a compact metric space is closed. – Brian M. Scott Feb 10 '21 at 21:47
  • Funny that you should mention explicitly that "compact implies bounded", but other instances such as "sequentially compact implies closed" are parcelised into "sequential compact implies compact implies closed". –  Feb 10 '21 at 22:05
  • @Gae.S. Oh that’s actually a great point. I imagine when I’m slightly more up for it I’ll update the diagram so that it doesn’t just include more arrows, it’ll also include specifics regarding the relationship each arrow represents, like applicability to sunsets of the metric space, etc, etc. – joshuaheckroodt Feb 10 '21 at 22:07
  • 1
    Even if the title does "speak for itself", you should make the body of your question self-contained. That also gives you more space to explain what you are asking. Here, it isn't very clear whether you are talking about properties of metric spaces in general or properties of subspaces of one metric space. – Rob Arthan Feb 10 '21 at 22:11
  • @RobArthan Done and done. – joshuaheckroodt Feb 10 '21 at 22:16
  • Great! Thank you. – Rob Arthan Feb 10 '21 at 22:18
  • I'm not sure how you would show it on your diagram, but a closed subspace of a compact metric space is compact. – Rob Arthan Feb 10 '21 at 22:19

0 Answers0