CW-Complex
A CW-complex is a homotopy-theoretic generalization of the notion of a Simplicial Complex. A CW-complex is anySpace 
which can be built by starting off with a discrete collection of points called 
, then attaching 1-DDisks 
to along their boundaries 
, writing 
for the object obtained by attaching thes to , then attaching 2-D Disks 
to along their boundaries 
, writing 
for thenew Space, and so on, giving spaces 
for every 
. A CW-complex is any Space that has this sort ofdecomposition into Subspaces built up in such a hierarchical fashion (so the s must exhaust allof ). In particular, may be built from 
by attaching infinitely many -Disks, and theattaching Maps 
may be any continuous Maps.
The main importance of CW-complexes is that, for the sake of Homotopy, Homology, and Cohomology groups, every Space is a CW-complex. This is called the CW-Approximation Theorem. Another is Whitehead's Theorem, which says that Maps between CW-complexes that induceIsomorphisms on all Homotopy Groups are actually Homotopy equivalences.
See also Cohomology, CW-Approximation Theorem, Homology Group, Homotopy Group, Simplicial Complex, Space, Subspace, Whitehead's Theorem- Map Folding
- Mapping (Function)
- Mapping Space
- Map Projection
- Marginal Analysis
- Marginal Probability
- Markoff Number
- Markoff's Formulas
- Markov Algorithm
- Markov Chain
- Markov Matrix
- Markov Moves
- Markov Number
- Markov Process
- Markov's Inequality
- Markov Spectrum
- Markov's Theorem
- Marriage Theorem
- Married Couples Problem
- Marshall-Edgeworth Index
- Martingale
- Mascheroni Constant
- Mascheroni Construction
- Maschke's Theorem
- Mason's abc Theorem