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cell attachment∗
yark†
2013-03-21 15:36:51
Let X be a topological space, and let Y be the adjunction Y := X ∪ϕ Dk ,
where Dk is a closed k-ball and ϕ : S k−1 → X is a continuous map, with S k−1
is the (k − 1)-sphere considered as the boundary of Dk . Then, we say that Y
is obtained from X by the attachment of a k-cell, by the attaching map ϕ. The
◦
image ek of Dk in Y is called a closed k-cell, and the image ek of the interior
D◦ := Dk \ S k−1
of Dk is the corresponding open k-cell.
Note that for k = 0 the above definition reduces to the statement that Y is
the disjoint union of X with a one-point space.
More generally, we say that Y is obtained from X by cell attachment if Y is
homeomorphic to an adjunction X ∪{ϕi } Dki , where
maps {ϕi } into X are
k the
i
defined on the boundary spheres of closed balls D .
∗ hCellAttachmenti created: h2013-03-21i by: hyarki version: h33991i Privacy setting:
h1i hDefinitioni h54B15i
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