Can a CW complex exist without being a Hausdorff space?

In summary, a CW complex is typically defined as a Hausdorff space. However, the question arises of whether a CW complex can exist without being Hausdorff. While some sources suggest that it may be possible, others argue that giving up the Hausdorff property is artificial and not clear what could be gained. Ultimately, a CW complex is automatically Hausdorff due to its definition involving open balls, and any two points in a CW complex will have disjoint open sets surrounding them, ensuring the Hausdorff property.
  • #1
viniciuslbo
4
0
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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  • #3
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.
 
  • #4
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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  • #5
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.
 
  • #6
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.
 
  • #7
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.)
 

1. What is a CW complex?

A CW complex is a type of topological space that is built by gluing together cells of different dimensions. It is named after mathematicians J.H.C. Whitehead and J.H.C. Moore, who developed the concept in the 1930s.

2. How is a CW complex different from other topological spaces?

Unlike other topological spaces, a CW complex is constructed inductively, starting with a 0-dimensional space (a set of points) and adding higher-dimensional cells one by one. This allows for a more flexible and intuitive understanding of the space.

3. What is the significance of the Hausdorff condition in a CW complex?

The Hausdorff condition, or the "Hausdorff separation axiom," requires that any two distinct points in a topological space can be separated by disjoint open sets. In a CW complex, this condition ensures that the cells are "well-behaved" and do not intersect in unexpected ways.

4. Can a CW complex have infinitely many cells?

Yes, a CW complex can have infinitely many cells, as long as it satisfies the Hausdorff condition. In fact, many important topological spaces, such as the real line and the surface of a sphere, can be constructed as CW complexes with an infinite number of cells.

5. What are some applications of CW complexes?

CW complexes have many applications in mathematics and physics, including in the study of homotopy theory, algebraic topology, and differential geometry. They are also used in modeling complex systems in science and engineering, such as in network theory and dynamical systems.

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