Can a CW complex exist without being a Hausdorff space?

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Discussion Overview

The discussion centers around the existence of CW complexes that are not Hausdorff spaces. Participants explore the implications of the Hausdorff condition in the context of CW complexes, examining definitions, properties, and potential consequences of relaxing this requirement.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant queries whether a CW complex can exist without being Hausdorff, suggesting that a cell decomposition of a non-Hausdorff space may not yield a 0-cell.
  • Several participants reference the Wikipedia definition of CW complexes, noting that it includes the Hausdorff condition as part of the definition.
  • Another participant expresses curiosity about what might be lost by abandoning the Hausdorff condition, indicating that this could be an interesting line of inquiry.
  • One participant argues that CW complexes are automatically Hausdorff, providing reasoning based on the properties of open balls and the subspace topology.
  • A later reply corrects a previous statement regarding the disjoint nature of open sets in the context of Hausdorff spaces.
  • Another participant reflects on the abstract nature of CW complexes, suggesting that giving up the Hausdorff condition seems artificial and questioning what benefits might arise from such a change.

Areas of Agreement / Disagreement

Participants express differing views on whether a CW complex can exist without being Hausdorff. While some assert that CW complexes are inherently Hausdorff, others question the implications of this condition and explore the theoretical aspects of non-Hausdorff spaces.

Contextual Notes

Participants reference definitions and properties from literature and Wikipedia, indicating that the discussion is influenced by existing definitions of CW complexes and their topological properties. There is an acknowledgment of the complexity surrounding the implications of the Hausdorff condition.

viniciuslbo
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I am with a query about cw complex. I was thinking if is possible exist a cw complex without being of Hausdorff space. Because i was thinking that when you do a cell decomposition of a space (without being of Hausdorff) you do not obtain a 0-cell. If can exist a cw complex with space without being of Hausdorff, someone can proof?
 
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mathman said:
https://en.wikipedia.org/wiki/CW_complex

I have no acquaintanceship with CW complex. However, the above description includes being Hausdorff as part of the definition.
That's what I've found in the books, too. Definitions start with it and is applied to the characteristic function where it is used.
However, the question what is lost by giving up Hausdorff could be interesting.
 
The Wiki page describes it as being Hausdorff : https://en.wikipedia.org/wiki/CW_complex

In addition, the restriction of the attaching map to the interior of the cell is a homeomorphism. But the cell, I assume is a Hausdorff space, and being Hausdorff is a topological property, so the image of the interior is Hausdorff. Now you need to deal with the image of the boundary, which " is contained in the union of a finite number of elements of the partition, each having cell dimension less than n."
 
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In the end it is an abstract concept that arose from geometry, Euclidean geometry. It's about triangulations, chain- and simplicial complexes. And cells which are little cubes. In this sense giving up Hausdorff is somehow artificial and it is not clear - at least to me - what could be gained.
 
CW complexes are automatically Hausdorff.

If X is a CW complex, X is the disjoint union of the interiors of at most countably infinitely many open balls Bk (a space homeomorphic to set of points in some Euclidean space whose distance from the origin is ≤ 1) such that each open ball retains its usually topology in the subspace topology.

Then any two points p, q of X with p ≠ q each lie in the interior of open balls Bi and Bk, respectively (where we cannot exclude the possibility that j = k).

Whether or not j = k, it follows that there exist open sets U ∋ p and V ∋ q of X, so X is Hausdorff.
 
In the last sentence in the above post, in my haste I omitted the word "disjoint" referring to U and V. It should read:

Whether or not j = k, it follows that there exist disjoint open sets U ∋ p and V ∋ q of X, so X is Hausdorff.

(Also: infinitely → infinitely.)
 

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