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cell attachment (Definition)

Let $ X$ be a topological space, and let $ Y$ be the adjunction $ Y := X\cup_\varphi D^k$, where $ D^k$ is a closed $ k$-ball and $ \varphi\colon S^{k-1}\rightarrow X$ is a continuous map, with $ S^{k-1}$ is the $ (k-1)$-sphere considered as the boundary of $ D^k$. Then, we say that $ Y$ is obtained from $ X$ by the attachment of a $ k$-cell, by the attaching map $ \varphi.$ The image $ e^k$ of $ D^k$ in $ Y$ is called a closed $ k$-cell, and the image $ \smash{\overset{\circ}e}^k$ of the interior

$\displaystyle D^\circ := D^k\setminus S^{k-1} $
of $ D^k$ 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\cup_{\left\{\varphi_i\right\}} D^{k_i}$, where the maps $ {\left\{\varphi_i\right\}}$ into $ X$ are defined on the boundary spheres of closed balls $ {\left\{D^{k_i}\right\}}$.



"cell attachment" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: CW complex

Other names:  cell adjunction
Also defines:  cell, open cell, closed cell, attaching map
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Cross-references: closed balls, spheres, maps, homeomorphic, disjoint union, interior, image, boundary, continuous map, adjunction, topological space
There are 34 references to this entry.

This is version 10 of cell attachment, born on 2003-02-07, modified 2007-05-20.
Object id is 3991, canonical name is CellAttachment.
Accessed 9095 times total.

Classification:
AMS MSC: 54B15 (General topology :: Basic constructions :: Quotient spaces, decompositions)
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