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Cellular approximation theorem
In algebraic topology, in the cellular approximation theorem, a map between CW-complexes can always be taken to be of a specific type. Concretely, if ''X'' and ''Y'' are CW-complexes, and ''f'' : ''X'' → ''Y'' is a continuous map, then ''f'' is said to be ''cellular'', if ''f'' takes the ''n''-skeleton of ''X'' to the ''n''-skeleton of ''Y'' for all ''n'', i.e. if for all ''n''. The content of the cellular approximation theorem is then that any continuous map ''f'' : ''X'' → ''Y'' between CW-complexes ''X'' and ''Y'' is homotopic to a cellular map, and if ''f'' is already cellular on a subcomplex ''A'' of ''X'', then we can furthermore choose the homotopy to be stationary on ''A''. From an algebraic topological viewpoint, any map between CW-complexes can thus be taken to be cellular.
== Idea of proof ==
The proof can be given by induction after ''n'', with the statement that ''f'' is cellular on the skeleton ''X''''n''. For the base case n=0, notice that every path-component of ''Y'' must contain a 0-cell. The image under ''f'' of a 0-cell of ''X'' can thus be connected to a 0-cell of ''Y'' by a path, but this gives a homotopy from ''f'' to a map which is cellular on the 0-skeleton of X.
Assume inductively that ''f'' is cellular on the (''n'' − 1)-skeleton of ''X'', and let ''e''''n'' be an ''n''-cell of ''X''. The closure of ''e''''n'' is compact in ''X'', being the image of the characteristic map of the cell, and hence the image of the closure of ''e''''n'' under ''f'' is also compact in ''Y''. Then it is a general result of CW-complexes that any compact subspace of a CW-complex meets (that is, intersects non-trivially) only finitely many cells of the complex. Thus ''f''(''e''''n'') meets at most finitely many cells of ''Y'', so we can take to be a cell of highest dimension meeting ''f''(''e''''n''). If , the map ''f'' is already cellular on ''e''''n'', since in this case only cells of the ''n''-skeleton of ''Y'' meets ''f''(''e''''n''), so we may assume that ''k'' > ''n''. It is then a technical, non-trivial result (see Hatcher) that the restriction of ''f'' to can be homotoped relative to ''X''''n-1'' to a map missing a point ''p'' ∈ ''e''''k''. Since ''Y''''k'' − deformation retracts onto the subspace ''Y''''k''-''e''''k'', we can further homotope the restriction of ''f'' to to a map, say, ''g'', with the property that ''g''(''e''''n'') misses the cell ''e''''k'' of ''Y'', still relative to ''X''''n-1''. Since ''f''(''e''''n'') met only finitely many cells of ''Y'' to begin with, we can repeat this process finitely many times to make miss all cells of ''Y'' of dimension larger than ''n''.
We repeat this process for every ''n''-cell of ''X'', fixing cells of the subcomplex ''A'' on which ''f'' is already cellular, and we thus obtain a homotopy (relative to the (''n'' − 1)-skeleton of ''X'' and the ''n''-cells of ''A'') of the restriction of ''f'' to ''X''''n'' to a map cellular on all cells of ''X'' of dimension at most ''n''. Using then the homotopy extension property to extend this to a homotopy on all of ''X'', and patching these homotopies together, will finish the proof. For details, consult Hatcher.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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