Baire Spaces and Dimension Theory

8 Baire Spaces and Dimension Theory

8.1 Baire Spaces

8.1.1 A has Empty interior ( a subset A of a space X) : A contains no open set of X other than the empty set.

8.1.1.1 A has empty interior if every point of A is a limit point of the complement of A.

8.1.1.2 The complement of A is dense in X.

8.1.2 Baire space ( a space X) : if the following condition holds.

8.1.2.1 Given any countable collection {A_{n}} of closed sets of X each of which has empty interior in X, their union \bigcup A_{n} also has empty interior in X.

8.1.3 First category in X ( a subset A of a space X) : it was contained in the union of a countable collection of closed sets of X having empty interiors in X.

8.1.3.1 Second category in X (a subset A of a space X) : A is not on the first category in X.

8.1.3.2 A space X is a Baire space iff every nonempty open set in X is of the second category.

8.1.4 (Lem 48.1) X is a Baire space iff given any countable collection {U_{n}} of open sets in X, each of which is dense in X, their intersection \bigcap U_{n} is also dense in X.

8.1.5 (Baire Category theorem) (Thm 48.2) : If X is a compact Hausdorff space or a complete metric space, then X is a Baire space.

8.1.6 (Lem 48.3) Let C_{1} \supset C_{2} \supset … be a nested sequence of nonempty closed sets in the complete metric space X. If diam C_{n} -> 0, then \bigcap C_{n} \neq \empty

8.1.7 (Lem 48.4) Any open subspace Y of a Baire space X is itself a Baire space.

8.1.8 (Thm 48.5) Let X be a space; let (Y,d) be a metric space. Let f_{n} : X -> Y be a sequence of continuous functions s.t. f_{n} (x) -> f(x) for all x \in X, where f : X-> Y. If X is a baire space, the set of points at which f is continuous is dense in X

8.1.8.1 Pf) (Lem 48.4)

8.2 Nowhere-Differentiable function

8.2.1 (Thm 49.1) Let h : [0,1] -> \mathbb{R} be a continuous function. Given \epsilon >0, there is a function g : [0,1] -> \mathbb{R} with \vert h(x) -> g(x) \vert < \epsilon for all x, s.t. g is continuous and nowhere differentiable.

8.2.1.1 Pf) (Lemma 48.1) (Complete metric space -> Baire space)

8.2.1.1.1 piecewise-linear (a function g) : a function whose graph is a broken line segment; each line segment in the graph of g has slope at least \alpha in absolute value

8.2.1.1.2 sawtooth graph

8.3 Introduction to Dimension Theory

8.3.1 \mathcal{A} has order m+1 ( a collection \mathcal{A} of subsets of the space X) : some points of X lies in m+1 elements of \mathcal{A}, and no point of X lies in more than m+1 elements of \mathcal{A}.

8.3.2 Finite dimensional ( a space X) : There is some integer m s.t. for every open covering \mathcal{A} of X, there is an open covering \mathcal{B} of X that refines \mathcal{A} and has order at most m+1.

8.3.3 Topological dimension of X (a space X) : the smallest value of m for a space X to be finite dimensional.

8.3.3.1 Dim X : Topological dimension of X

8.3.3.2 Example

8.3.3.2.1 Any compact subspace X of \mathcal{R} has topological dimension at most 1.

8.3.3.2.2 Any compact subspace X of \mathcal{R}^{2} has topological dimension at most 2.

8.3.4 (Thm 50.1) Let X be a space having finite dimension. If Y is a closed subspace of X, then Y has finite dimension and dim Y \le dim X.

8.3.5 (Thm 50.2) Let X = Y \cup Z, where Y and Z are closed subspaces of X having finite topological dimension. Then dim X = max {dim Y, dim Z}

8.3.5.1 Pf) (Thm 50.1)

8.3.5.2 (Cor 50.3) Let X = Y_{1} \cup … \cup Y_{k}, where each Y_{i} is a closed subspace of X and is finite dimensional. Then dim X = max{dim Y_{1} , .., dim Y_{k}}.

8.3.6 Arc : a space homeomorphic to the closed unit interval

8.3.6.1 Endpoints of Arc : the points p and q s.t. A – {p} and A-{q} are connected.

8.3.6.2 Linear graph G : a Hausdorff space that is written as the union of finitely many arcs, each pair of which intersects in at most a common end point.

8.3.6.3 Edges of G : arcs in the collection

8.3.6.4 Vertices of G : endpoints of arcs

8.3.6.5 Each edge of G is closed in G; G has topological dimension 1.

8.3.6.6 Every finite linear graph can be imbedded in \mathbb{R}^{3}

8.3.7 Geometrically independent (a set {\mathbf{x}{0},…, \mathbf{x}{k}} of points of \mathbb{R}^{N} ) : if the equation \sum_{I = 0}^{k} a_{i}mathbf{x}{i} = \mathbf{0} and \sum{I = 0}^{k} a_{i} = 0 holds only if each a_{i} = 0.

8.3.8 Plane P determined by these points (a set of all points \mathbf{x} of of \mathbb{R}^{N} ) : Let {\mathbf{x}{0},…, \mathbf{x}{k}} be a set of points of \mathbb{R}^{N} that is geometrically independent.\mathbf{x} = \sum_{I = 0}^{k} t_{i}mathbf{x}{i} where \sum{I = 0}^{k} t_{i} = 1.

8.3.8.1 mathbf{x} = \mathbf{x}{0} + \sum{I = 0}^{k} a_{i}( mathbf{x}{i} - \mathbf{x}{0}) for some scalars a_{1} , …, a_{k}

8.3.8.2 Translation T of \mathbb{R}^{N} : a homeomorphism T : \mathbb{R}^{N} -> \mathbb{R}^{N} ; T(\mathbf{x}) = \mathbf{x – x_{0}}

8.3.8.3 K-plane P in \mathbb{R}^{N} : a plane P

8.3.9 A in general position in \mathbb{R}^{N} (a set A of points of \mathbb{R}^{N}) : every subset of A containing N + 1 or fewer points is geometrically independent

8.3.10 (Lem 50.4) Given a finite set {\mathbf{x}{1},…, \mathbf{x}{n}} of points of \mathbb{R}^{N} and given \delta >0, there exists a set {\mathbf{y}{1},…, \mathbf{y}{n}} of points of \mathbb{R}^{N} in general position in \mathbb{R}^{N}, s.t. \vert x_{i} – y_{i} \vert < \delta for all i.

8.3.10.1 Pf) (Def of Baire space)

8.3.11 (The imbedding Theorem) (Thm 50.5) Every compact metrizable space X of topological dimension m can be imbedded in \mathbb{R}^{2m+1}

8.3.12 (Thm 50.6) Every compact subspace of \mathbb{R}^{N} has topological dimension at most N.

8.3.12.1 Pf) M-cube : homeomorphic to the product (0,1)^{M}

8.3.12.2 (Cor 50.7) Every compact m-manifold has topological dimension at most m.

8.3.12.3 (Cor 50.8) Every compact m-manifold con be imbedded in \mathbb{R}^{2m+1}

8.3.12.4 (Cor 50.9) Let X be a compact metrizable space. Then X can be imbedded in some Euclidean space \mathbb{R}^{N} iff X has finite topological dimension.