Every open subset of R^p is the union of countable collection of

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Homework Help Overview

The discussion revolves around the properties of open subsets in R^p, specifically focusing on their representation as unions of countable collections of closed sets. Participants are exploring the implications of this property in the context of topology.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to seek guidance on progressing with their understanding of the problem. Some participants question the relationship between irrational and rational numbers within open balls, while others note the equivalence to R^p being 2nd-countable.

Discussion Status

The discussion is active, with participants providing insights and raising questions. Some guidance has been offered regarding the properties of rational numbers in open sets, and there is acknowledgment of the equivalence to 2nd-countability. However, there is no explicit consensus on the overall approach yet.

Contextual Notes

The original poster has attached an image of their attempt, indicating a potential constraint in conveying their reasoning through text alone. The nature of the problem suggests a focus on foundational concepts in topology.

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Every open sub set of Rp is the union of countable collection of closed sets.

I am attaching my attempt as an image file. Please guide me on how I should move ahead. Thank you very much for your help.
 

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Take an irrational number ##a## in ##G##. You can find a certain open ball ##B(a,\varepsilon)## which remains in ##G##. What can you tell about the rational numbers in that ball? In particular, if ##q\in B(a,\varepsilon)## is rational, what can you tell about the closed set in the hint?
 
This is equivalent to R^p being 2nd-countable.
 
I think I got it. Thank you for your comments
 

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