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4
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'cell attachment'
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| Title of object: |
cell attachment |
| Canonical Name: |
CellAttachment |
| Type: |
Definition |
| Created on: |
2003-02-07 13:58:31.939214-05 |
| Modified on: |
2003-02-07 17:50:00.860246-05 |
| Classification: |
msc:54B15 |
| Defines: |
cell, open cell, closed cell |
| Synonyms: |
cell attachment=cell adjunction |
Revision comment (for changes between this and next version):
| got rid of the remark, which wasn't quite right. i might add a correct version later. |
Preamble:
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Content:
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 $\funcdef{\varphi}{S^{k-1}}{X}$ 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 {\em attachment of a $k$-cell.}
The image $e^k$ of $D^k$ in $Y$ is called a {\em closed $k$-cell,} and the image $\oce^k$ of the interior $\ocD := D^k\setminus S^{k-1}$ of $D^k$ is the corresponding {\em 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 {\em cell attachment\/}
if $Y$ is homeomorphic to an adjunction $X\cup_\set{\varphi_i} D^{k_i},$ where the maps $\set{\varphi_i}$ into $X$ are defined on the boundary spheres of closed balls $\set{D^{k_i}}.$
\begin{rmk}
A recognition principle for attached cells is as follows: Let $Y$ be a Hausdorff topological space and $e$ a closed subspace such that there exists a map $\funcdef{\Phi}{D^k}{Y}, k\ge 1,$ satisfying:
\begin{enumerate}
\item
$\Phi(D^k) = e$ and
\item
the restriction of $\Phi$ to $\ocD^k := D^k\setminus \bdry D^k$ is an embedding.
\end{enumerate}
Then, $Y$ is obtained from $X := Y\setminus\Phi(\ocD^k)$ by the attachment of the $k$-cell $e$. $k$ is called the {\em dimension} of $e,$ and is well-defined by virtue of the invariance of domain theorem.
Attached $0$-cells are recognized as being isolated points of $X$.
\end{rmk} |
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