Large inductive dimension: Difference between revisions

Revision as of 22:46, 10 November 2007

Definition

The large inductive dimension of a topological space is defined inductively. The empty set is assigned dimension $- 1$ . Suppose we have defined what it means for a topological space to have dimension $\leq m$ . Then a topological space $X$ has dimension $\leq m + 1$ if given any two closed subsets $A, B \subset X$ , there exists a closed subset $P \subset X$ of dimension $\leq m$ such that the complement of $P$ is a disjoint union of open sets $C$ and $D$ where $C$ contains $A$ and $D$ contains $B$ .

The large inductive dimension of $X$ is denoted $I n d X$ .

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 The large inductive dimension of a topological space is defined inductively. The empty set is assigned dimension <math>-1</math>. Suppose we have defined what it means for a topological space to have dimension <math>\le m</math>. Then a topological space <math>X</math> has dimension <math>\le m + 1</math> if given any two closed subsets <math>A, B \subset X</math>, there exists a closed subset <math>P \subset X</math> of dimension <math>\le m</math> such that the complement of <math>P</math> is a disjoint union of open sets <math>C</math> and <math>D</math> where <math>C</math> contains <math>A</math> and <math>D</math> contains <math>B</math>.
-The large inductive dimension of <math>X</math> is denoted <math>Ind X</math>.
+The large inductive dimension of <math>X</math> is denoted <math>Ind  \ X</math>.
 ==Related notions==