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For these purposes a closed cell is a topological space homeomorphic to a , for some equivalence relation. ATTACHING CELLS A cell is Attached by gluing a closed ''n''-dimensional ball ''D''''n'' to the (''n''−1)-''skeleton'' ''X''''n''−1, i.e., the union of all lower dimensional cells. The gluing is specified by a continuous function ''f'' from ∂''D''''n'' = ''S''''n''−1 to ''X''''n''−1. The points on the new space are exactly the equivalence classes of points in the disjoint union of the old space and the closed cell ''D''''n'', the equivalence relation being the Transitive Closure of ''x'' ≡ ''f''(''x''). The function ''f'' plays an essential role in determining the nature of the newly enlarged complex. For example, if ''D''2 is glued onto ''S''1 in the usual way, we get ''D''2 itself; if ''f'' has Winding Number 2, we get the Real Projective Plane instead. Regular CW-complex If all attaching maps are Homeomorphisms , the structure is called a ''regular CW-complex''. CW COMPLEXES ARE DEFINED INDUCTIVELY Assume that ''X'' to be a s, we can be sure that their images in ''X'' are also compact, closed subspaces. From now on, we refer to 'closed cells', and 'open cells', as subspaces of ''X'', the open cell being the image of the distinguished interior. A 0-cell is just a point; if we only have 0-cells building up a Hausdorff space, it must be a Discrete Space . The general CW-complex definition can proceed by induction, using this as the base case. The first restriction is the closure-finite one: each closed cell should be covered by a Finite Union of open cells. The other restriction is to do with the possibility of having infinitely many cells, of unbounded dimension. The space ''X'' will be presented as a limit of subspaces ''X''''i'' for ''i'' = 0, 1, 2, 3, … . How do we infer a topological structure for X? This is a Colimit in Category Theory terms. From the continuity of each mapping ''X''''i'' to ''X'', a closed set in ''X'' must have a closed inverse image in each ''X''''i'', and so must intersect each closed cell in a closed subset. We can turn this round, and require that a subset ''C'' ⊂ ''X'' is by definition closed precisely when the intersection of ''C'' with the closed cells in ''X'' is always closed. This yields the weak topology on ''X''. With all those preliminaries, the definition of CW-complex runs like this: given ''X''0 a discrete space, and inductively constructed subspaces ''X''''i'' obtained from ''X''''i''−1 by attaching some collection of ''i''-cells, the resulting colimit space ''X'' is called a CW-complex provided it is given the weak topology, and the closure-finite condition is satisfied for its closed cells. 'THE' HOMOTOPY CATEGORY The Homotopy Category of CW complexes is, in the opinion of some experts, the best if not the only candidate for ''the'' homotopy category. Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion, but half a century since Whitehead has left this definition of homotopy category in good shape. One basic result is that the Representable Functor s on the homotopy category have a simple characterisation (the Brown Representability Theorem ). PROPERTIES
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