Section 36*: Imbeddings of Manifolds

1. An m-manifold is a Hausdorff second-countable space such that every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^m$ .
2. being Hausdorff is not a local property, and without requiring it an m-manifold does need to be Hausdorff; however, being a $T_1$ -space is a local property, and, in fact, if we did not require a manifold to be a Hausdorff, it would be at least $T_1$
3. second-countability requirement is needed for the theorem that states that it can be imbedded into an euclidean space (obviously, both the Hausdorff property and second-countability are necessary, but they turn out to be sufficient): see below
4. if the space is compact and satisfies all the properties in the definition of an $m$ -manifold except for second-countability then it must be second-countable and, hence, $m$ -manifold: see below
5. a manifold is regular (given that it is Hausdorff), therefore, metrizable (being regular and second-countable)
6. The support of $\phi:X\to\mathbb{R}$ is the closure of $\phi^{-1}(\mathbb{R}-\{0\})$ .
7. A point-finite collection of subsets is a collection of subsets such that every point is contained in a finite number of sets in the collection.
8. Compare to a stronger notion of a locally finite collection of subsets (Section 17).
9. A partition of unity: a set of functions $\{\rho_j:X\to[0,1]\}$ such that for every $x\in X$ only a finite number of $\rho_j(x)\neq 0$ and the sum of all $\rho_j(x)$ equals $1$ .
10. A partition of unity is said to be dominated by an open covering$\{U_j\}$ indexed over the same index set if for every $j$ , the support of $\rho_j$ is contained in $U_j$ .
11. If $X$ is normal then for every finite collection of open sets covering $X$ there is a partition of unity dominated by the open covering.
12. In fact, the result holds true for every countable point-finite collection of open sets covering the normal space.
13. If space is a $m$ -manifold then it can be imbedded in a $n$ -dimensional euclidean space where $n\in\mathbb{Z}_+$ .
14. This chapter proves the result for compact spaces only.
15. We can cover the space by a finite collection of n open sets such that each is homeomorphic to the m-dimensional euclidean space, then find a partition of unity dominated by the collection; then we map each point to a point in the (n+nm)-dimensional euclidean space by taking the values at the point of the functions in the partition of unity and also by taking their products with the homeomorphisms (or zero if the homeomorphism is not defined for the point — this is why we need to take the product, so that the resulting function is continuous); this mapping is continuous and injective (the partition of unity part of the resulting vector ensures that points not in the same open set have different coordinates, while the other part results in different coordinates for points in the support of the same function in the partition of unity).
16. Compact Hausdorff spaces and manifolds: let $X$ be compact and Hausdorff, then
17. If each point $x$ has a neighborhood that can be imbedded into $\mathbb{R}^{k_x}$ for some $k_x$ then $X$ can be imbedded into some finite-dimensional euclidean space as well.
18. If for some $m$ each point $x$ has a neighborhood homeomorphic to an open subset of $\mathbb{R}^m$ then $X$ is an $m$ -manifold.