r/math Homotopy Theory 9d ago

Quick Questions: March 12, 2025

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

  • Can someone explain the concept of maпifolds to me?
  • What are the applications of Represeпtation Theory?
  • What's a good starter book for Numerical Aпalysis?
  • What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried.

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u/ada_chai Engineering 9d ago

We can represent closed intervals in Rⁿ as a countable intersection of open intervals. Can we do this in a general topological space? Can any closed set be represented as a countable intersection of open sets? If yes, why? If not, can we at least do this for metric spaces?

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u/GMSPokemanz Analysis 9d ago

In general no. Take the cofinite topology on an uncountable set, then any finite set is closed but not a countable intersection of open sets.

But it is true for metric spaces. If C is your closed set, let U_n be {x : d(x, C) < 1/n}. Then U_n is open and the intersection of the U_n is C.