# CW Complex

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A CW complex is an area containing several very small buildings (called cells). Cells are usually Cobbleboxes, but sometimes they feature windows. A constructor of CW complexes is called a **topologist**.

The difference between a CW complex and a village is mostly the size of the buildings. Villages have nice houses, CW complexes are composed of Cells.

The construction of CW complexes has been mastered by morgan8 who produced the Morganian Style of Architecture by crossing CW complex construction with Osthoffian Style Architecture.

## Contents

## Etymology

It comes from the mathematical of a CW complex, a certain kind of "nice" topological space, where the "C" stands for "Cells", hence the term *topologist* for a CW complex builder.

## Construction

A CW Complex X is composed of a 0-skeleton X^{0}, which is just a discrete set of nodes, each of which represents a corner of the CW complex' cells. To X^{0}, then the 1-skeleton X^{1} is glued, which is composed of 1-cells (edges of the cells). To X^{1}, we then glue the 2-skeleton X^{2}, which is composed of the walls of the CW complex. The complex is then the union of the 0,1 and 2 skeletons of the complex.

## Homology

One of the key properties of a CW complex is its *homology*. Homology can roughly be described as an abelian group which describes the number and dimension of holes in the space. Two approaches for homology of CW complexes are knouwn. Gabrielian Homology, which does not include holes in bars or doors; and Humean Homology, which includes those holes

### Gabrielian Homology

This homology only counts windows and doors as holes. It is seen as a very practical kind of homology, since it is easy to compute and is material-independent, it gives the same results regardless of the kind of door material used.

### Humean Homology

Created by User:Hume2, this homology counts all holes.

#### Criticism

User:gabriel, creator of Gabrielian Homology criticizes that this kind of homology counts irrelevant holes, since holes in iron bars or doors are seen as irrelevant for the function of the CW complex.

### Interpretation of Homology

The H_{2}(X) Homology group gives information about the number of 2-cells (also called simply cells or sometimes sarcastically "rooms"), the H_{1}(X) group gives info about the number of windows and doors. CW Complexes in which the H_{2} group is not significantly smaller than the H_{1} group are called *under-windowed* or *dungeonlike*, as this indicates a significant lack of windows. This shows one of the big advantages of Gabrielian homology, since the topologist constructing the CW complex cannot simply cheat and claim to have a huge number of windows by simply using wooden doors and iron bars as windows.