Problem of the Week #109 - June 30th, 2014

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SUMMARY

The discussion centers on the mathematical problem of determining whether every locally compact Hausdorff space is paracompact. It is established that not all locally compact Hausdorff spaces are paracompact, with the Tychonoff plank serving as a notable counterexample. The problem remains unsolved in the forum, with participants encouraged to explore this topic further.

PREREQUISITES
  • Understanding of locally compact Hausdorff spaces
  • Familiarity with paracompactness in topology
  • Knowledge of counterexamples in mathematical proofs
  • Basic concepts of topology and set theory
NEXT STEPS
  • Research the properties of the Tychonoff plank as a counterexample
  • Study the definitions and implications of paracompactness
  • Explore the relationship between compactness and paracompactness
  • Investigate other examples of locally compact spaces
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in topology, as well as students and educators seeking to deepen their understanding of compactness and paracompactness in mathematical spaces.

Chris L T521
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Here's this week's problem!

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Problem: Determine whether or not every locally compact Hausdorff space is paracompact.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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No one answered this week's problem.

I don't have a complete solution prepared due to various things, so I will do my best to fill in the details sometime soon.

[sp]It turns out that not every locally compact Hausdorff space is paracompact. An interesting counterexample is the Tychonoff plank, which I will elaborate on soon.[/sp]
 

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