# Three things I don't get about CW-complexes

• quasar987
In summary, a CW-complex is a type of topological space made up of cells and attaching maps. It has several advantages, such as flexibility and a nice combinatorial structure, and can be used to decompose a space into simpler pieces. It differs from simplicial complexes in its construction and has key properties like being Hausdorff and locally contractible. The "CW" in CW-complex refers to its closure-finite and weak topology construction.

#### quasar987

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(1) We say a CW-complex is finite if its cell decompositions are finite. Why is this well defined? I.e., why can't a CW-complex admits a decomposition into a finite number of cells and another decomposition into an infinite number of cells?

(2) We say a CW-complex X has dimension n if n is the dimension of the cell of X of highest dimension. Why is this well defined? I.e., why can't a CW-complex admits a cell decomposition where the highest dimensional cell has dimension n and another cell decomposition where the highest dimensional cell has dimension m? I'm guessing this can be shown using Brouwer's invariance of domain, but I've been having technical problems implementing this idea..

(3) Given a CW-complex X, a subcomplex of X can be defined as a subspace A of X which is the reunion of closed cells of X. Why is a subcomplex necessarily closed in X?

Thanks for any help.

quasar987 said:
(1) We say a CW-complex is finite if its cell decompositions are finite. Why is this well defined? I.e., why can't a CW-complex admits a decomposition into a finite number of cells and another decomposition into an infinite number of cells?

(2) We say a CW-complex X has dimension n if n is the dimension of the cell of X of highest dimension. Why is this well defined? I.e., why can't a CW-complex admits a cell decomposition where the highest dimensional cell has dimension n and another cell decomposition where the highest dimensional cell has dimension m? I'm guessing this can be shown using Brouwer's invariance of domain, but I've been having technical problems implementing this idea..

(3) Given a CW-complex X, a subcomplex of X can be defined as a subspace A of X which is the reunion of closed cells of X. Why is a subcomplex necessarily closed in X?

Thanks for any help.

1) A finite CW complex is compact An infinite one is not.

2) In the top dimension the CW complex contains open sets homeomorphic to R_n not R_n+1.

3) by definition the closure of each open cell in a sub-complex is contained in the sub-complex

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## 1. What is a CW-complex?

A CW-complex is a type of topological space that is commonly used to study the structure of a given space. It is made up of a collection of cells, which are topological building blocks, and their attaching maps. These cells are usually labeled by their dimension and are attached to each other in a specific way to form the CW-complex.

## 2. What are the advantages of using CW-complexes?

CW-complexes have several advantages in studying topological spaces. They are flexible and can be used to construct a wide variety of spaces, including familiar ones like spheres and tori. They also have a nice combinatorial structure, which makes them easier to work with and understand. Additionally, they provide a way to decompose a space into simpler pieces, which can aid in understanding its topology.

## 3. How are CW-complexes different from simplicial complexes?

CW-complexes and simplicial complexes are both types of topological spaces, but they differ in their construction. While a simplicial complex is made up of simplices (geometric objects with flat sides), a CW-complex is made up of cells (which can have more complicated shapes). Additionally, cells in a CW-complex are attached to each other in a specific way, while simplices in a simplicial complex are simply connected at their vertices.

## 4. What are the key properties of CW-complexes?

CW-complexes have several key properties that make them useful in topology. Firstly, they are Hausdorff spaces, meaning that any two points in the space can be separated by disjoint open sets. Secondly, they are locally contractible, which means that they are "smooth" enough to do calculus on. Finally, they have a finite number of cells in each dimension, which can help in understanding their structure.

## 5. What is the significance of the "CW" in CW-complexes?

The "CW" in CW-complex stands for "closure-finite" and "weak topology." These terms refer to the way that cells are attached to each other in a CW-complex. "Closure-finite" means that the closure of each cell is contained in the union of cells of lower dimension. "Weak topology" refers to the fact that the topology of a CW-complex is determined by its open cells, rather than its closed cells.