125 we shouldn't force the linear indexing of the blobs to have anything to do with |
125 we shouldn't force the linear indexing of the blobs to have anything to do with |
126 the partial ordering by inclusion -- this is what happens below} |
126 the partial ordering by inclusion -- this is what happens below} |
127 \nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} |
127 \nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} |
128 } |
128 } |
129 |
129 |
|
130 \begin{defn} |
|
131 An \emph{$n$-ball decomposition} of a topological space $X$ is |
|
132 finite collection of triples $\{(B_i, X_i, Y_i)\}_{i=1,\ldots, k}$ where $B_i$ is an $n$-ball, $X_i$ is some topological space, and $Y_i$ is pair of disjoint homeomorphic $n-1$-manifolds in the boundary of $X_{i-1}$ (for convenience, $X_0 = Y_1 = \eset$), such that $X_{i+1} = X_i \cup_{Y_i} B_i$ and $X = X_k \cup_{Y_k} B_k$. |
|
133 |
|
134 Equivalently, we can define a ball decomposition inductively. A ball decomposition of $X$ is a topological space $X'$ along with a pair of disjoint homeomorphic $n-1$-manifolds $Y \subset \bdy X$, so $X = X' \bigcup_Y \selfarrow$, and $X'$ is either a disjoint union of balls, or a topological space equipped with a ball decomposition. |
|
135 \end{defn} |
|
136 |
|
137 |
130 Before describing the general case we should say more precisely what we mean by |
138 Before describing the general case we should say more precisely what we mean by |
131 disjoint and nested blobs. |
139 disjoint and nested blobs. |
132 Disjoint will mean disjoint interiors. |
140 Disjoint will mean disjoint interiors. |
133 Nested blobs are allowed to coincide, or to have overlapping boundaries. |
141 Nested blobs are allowed to coincide, or to have overlapping boundaries. |
134 Blob are allowed to intersect $\bd X$. |
142 Blob are allowed to intersect $\bd X$. |