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... If we require g ◦ f = 0, i.e., Im(f ) ⊂ ker(g), then we have a chain complex and homology groups are defined. We say the sequence is exact (at B) if we have ker(g) = Im(f ). This condition is to equivalent to saying the homology group at B is trivial, and thus homology groups of a chain complex meas ...
... If we require g ◦ f = 0, i.e., Im(f ) ⊂ ker(g), then we have a chain complex and homology groups are defined. We say the sequence is exact (at B) if we have ker(g) = Im(f ). This condition is to equivalent to saying the homology group at B is trivial, and thus homology groups of a chain complex meas ...
Garrett 12-14-2011 1 Interlude/preview: Fourier analysis on Q
... b an abelian group. A reasonable topology on G b is the compactG open topology, with a sub-basis b : f (C) ⊂ E} U = UC,E = {f ∈ G for compact C ⊂ G, open E ⊂ S 1 . Remark: The reasonable-ness of this topology is functional. For a compact topological space X, C o (X) with the sup-norm is a Banach spa ...
... b an abelian group. A reasonable topology on G b is the compactG open topology, with a sub-basis b : f (C) ⊂ E} U = UC,E = {f ∈ G for compact C ⊂ G, open E ⊂ S 1 . Remark: The reasonable-ness of this topology is functional. For a compact topological space X, C o (X) with the sup-norm is a Banach spa ...
WHEN ARE ALL CONTINUOUS FUNCTIONS TO Y
... to Y is constant on some neighborhood of Ri . Then we show that the space T , which is obtained from the product R1 × R2 by removing the point (r1 ; r2 ), has the property, that for every continuous map f from T to Y there exist neighborhoods Ui of ri such that f is constant on U1 × U2 − {(r1 ; r2 ) ...
... to Y is constant on some neighborhood of Ri . Then we show that the space T , which is obtained from the product R1 × R2 by removing the point (r1 ; r2 ), has the property, that for every continuous map f from T to Y there exist neighborhoods Ui of ri such that f is constant on U1 × U2 − {(r1 ; r2 ) ...
