The realization of abstract stratified sets
Hiroko Natsume · Kodai Mathematical Journal · 1980
It is well-known that n-dimensional differentiable manifolds can be realized in the (2n + l)-dimensional Euclidean space.Recently, R. Thorn and J. Mather have introduced the notion of abstract stratified set, modelling after variety (manifold with singuralities).In this paper, we shall realize n-dimensional stratified sets in the (2n + l)-dimensional Euclidean space. DEFINITION 1 ([1]).A stratification S for a subset V in R N is a locally finite family of pairwise disjoint submanifolds X of R N , satisfying the following conditions.51.Each X^S lies in V, which is called a stratum of {V, S}. 52.Each point of V is contained in the interior of some stratum.53.The frontier condition: for each stratum X, if a stratum Y intersects with the closure X in V of X, then YdX.If YaX, Y is said to be incident to X and we write Y Y.A topological space V with a stratification S is called a stratified set.H. Whitney in [1] considered stratified sets with the following condition. Whitney condition.For each pair of strata (X, Y) such that X>Y, if both series of points {x x } in X and {y^ in Y converge to a point y in Y and the line through x x and y x converges to some line / and the tangent space of X at x % converges to some plane P, then iczP.R. Thorn ([3]) and J. Mather ([2]), axiomatizing stratified set together with a tubular neighbourhood system, introduced the following notion of abstract stratified sets.DEFINITION 2 ([2]).Let V be a subset of R N with a stratification S. A family £Γ-{(T x , π Xί pχ)}χ^s is called a tubular neighbourhood system for S, if it satisfies the following conditions:is a bundle such that there exists a bundle isomorphism ψx'-T x -^Bχ where B x is the open unit ball bundle in an inner product bundle over X.