Connectedness and Compactness

3 Connectedness and Compactness

3.1 Connected spaces :

3.1.1 Separation of X ( a topological space X) : a pair U, V of disjoint nonempty open subsets of X whose union is X.

3.1.2 Connected ( a topological space X) if there does not exist a separation of X.

3.1.2.1 The only subsets of X that are both open and closed in X are the empty set and X itself.

3.1.3 (Lem 23.1) If Y is a subspace of X, a separation of Y is a pair of disjoint nonempty sets A and B whose union is Y, neither of which contains a limit point of the other. The space Y is connected if there exists no separation of Y.

3.1.4 (Lem 23.2) If the sets C and D form a separation of X, and if Y is a connected subspace of X then Y lies entirely within either C or D.

3.1.5 (Thm 23.3) The union of a collection of connected subspaces of X that have a point in common is connected.

3.1.6 (Thm 23.4) Let A be a connected subspace of X. If A \subset B \subset \bar{A}, then B is also connected.

3.1.6.1 Pf) Lem 23.2

3.1.7 (Thm 23.5) The image of a connected space under a continuous map is connected.

3.1.8 (Thm 23.6) A finite cartesian product of connected spaces is connected.

3.1.9 An arbitrary product of connected spaces is connected in the product topology.

3.2 Connected subspaces of the real line.

3.2.1 Linear continnum ( a simply ordered set L) : if following holds.

3.2.1.1 L has the least upper bound property.

3.2.1.2 If x < y, there exists z s.t. x < z < y.

3.2.2 (Thm 24.1) If L is a linear continuum in the order topology, then L is connected, and so are intervals and rays in L.

3.2.2.1 Pf) convex

3.2.2.2 (Cor) The real line \mathbb{R} is connected and so are intervals and rays in \mathbb{R}.

3.2.3 (Intermediate value theorem ) (Thm 24.3): Let f : X -> Y be a continuous map, where X is a connected space and Y is an ordered set in the oerder topology. If a and b are two points of X and if r is a point of Y lying between f(a) and f(b), then there exists a point c of X s.t. f(c) = r

3.2.4 Path in X from x to Y ( points x, y of the space X) : continuous map f : [a,b] -> X of some closed interval in the real line into X, s.t. f(a) = x and f(b) = y.

3.2.5 Path connected (a space X) : every pair of points of X can be joined by a path in X.

3.2.5.1 Path connected space X is connected.

3.2.6 Examples

3.2.6.1 Unit ball B^{n} in \mathbb{R}^{n} : B^{n} = {\mathbf{x} | \Vert x \Vert \le 1}

3.2.6.1.1 \Vert \mathbf{x} \Vert = \Vert (x_{1}, …, x_{n}) \Vert = (x^{2}{1} + .. + x^{2}{n} )^{fact{1}{2}}

3.2.6.1.2 Unit ball is path connected.

3.2.6.2 Punctured Euclidean space : \mathbb{R}^{n} – {\mathbf{0}}

3.2.6.2.1 \mathbf{0} : origin of \mathbb{R}^{n}.

3.2.6.2.2 If n > 1, it is path connected.

3.2.6.3 Unit sphere S^{n-1} in \mathbb{R}^{n} : S^{n-1} = {\mathbf{x} | \Vert \mathbf{x} \Vert = 1}

3.2.6.3.1 If n > 1, it is path connected

3.2.6.4 Ordered square I^{2}_{0} is connected but not path connected.

3.2.6.5 Topologist’s sine curve : S = {x \times sin(1/x) | 0<x \le 1}

3.2.6.5.1 Connected but not path connected

3.3 Components and Local connectedness

3.3.1 components of X (a space X) : Equivalent classes where an equivalence relation on X by setting x ~ y if there is a connected subspace of X containing both x and y.

3.3.1.1 Also called connected components

3.3.2 (Thm 25.1) The components of X are connected disjoint subspaces of X whose union is X, s.t. each nonempty connected subspace of X intersects only one of them.

3.3.3 Path components of X (a space X) : Equivalent classes where an equivalence relation on X by defining x ~ y if there is a path in X from x to y.

3.3.3.1 Pf ) (pasting lemma)

3.3.4 (Thm 25.2) The path components of X are path-connected disjoint subspaces of X whose union is X, s.t. each nonempty path connected subspace of X intersects only one of them.

3.3.5 Locally connected at x (a point x in a space X) : For every neighborhood U of x, there is a connected neighborhood V of x contained by U.

3.3.6 Locally connected (a space X): X is locally connected at each of its points.

3.3.7 Locally path connected at x (a point x in a space X) : For every neighborhood U of x, there is a path connected neighborhood V of x contained by U.

3.3.8 Locally path connected (a space X): X is locally path connected at each of its points.

3.3.9 (Thm 25.3) A space X is locally connected iff for every open set U of X, each component of U is open in X.

3.3.10 (Thm 25.4) A space X is locally path connected iff for every open set U of X, each path component of U is open in X.

3.3.11 (Thm 25.5) If X is a topological space, each path component of X lies in a component of X. If X is a locally path connected, then the components and the path components of X are the same.

3.4 Compact spaces

3.4.1 Covering of X (a collection \mathcal{A} of subsets of a space X) : the union of the elements of \mathcal{A} is equal to X.

3.4.1.1 Also called ‘to cover X’.

3.4.1.2 Open covering of X if its elements are open subsets of X.

3.4.2 Compact (a space X) : every open covering \mathcal{A} of X contains a finite subcollection that also covers X

3.4.3 Cover Y (a subspace Y of X) : If, for a collection \mathcal{A} of subsets of X, the union of its elements contains Y.

3.4.4 (Lem 26.1) Let Y be a subspace of X. Then Y is compact iff every covering of Y by sets open in X contains a finite subcollection covering Y.

3.4.5 (Thm 26.2) Every closed subspace of a compact space is compact.

3.4.6 (Thm 26.3) Every compact subspace of a Hausdorff space is closed.

3.4.7 (Lem 26.4) If Y is a compact subspace of the Hausdorff space X and x_{0} is not in Y, then there exist disjoint open sets U and V of X containing x_{0} and Y, tespectively.

3.4.8 (Thm 26.5) The image of a compact space under a continuous map is compact.

3.4.9 (Thm 26.6) Let f : X -> Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism.

3.4.9.1 (Thm 26.2) (Thm 26.5) (Thm 26.3)

3.4.10 (Thm 26.7) The product of finitely many compact spaces is compact)

3.4.10.1 (Tube Lemma) (Lem 26.8) : Consider the product space X \times Y, where Y is compact. If N is an open set of X \times Y containing the slice x_{0} \times Y of X \times Y, then N contains some tube W \times Y about x_{0} \times Y, where W is a neighborhood of x_{0} in X.

3.4.11 Finite intersection property (a collection \mathcal{C} of subsets of X) : For every finite subcollection {C_{1}, …, C_{n}} of \mathcal{C}, the intersection C_{1} \cap … \cap C_{n} is nonempty.

3.4.12 (Thm 26.9) Let X be a topological space. Then X is compact iff for every collection \mathcal{C} of closed sets in X having the finite intersection property, the intersection \bigcup_{C \in \mathcal{C}} C of all the elements of \mathcal{C} is nonempty.

3.4.12.1 Nested sequence C_{1} \supset C_{2} \supset … \supset C_{n} \supset .. of closed sets in a compact space X

3.5 Compact subspaces of the Real line

3.5.1 (Thm 27.1) Let X be a simply ordered set having the least upper bound property. In the order topology, each closed interval in X is compact.

3.5.1.1 Pf) Subspace topology equals order topology

3.5.1.2 (Cor) Every closed interval in \mathbb{R} is compact.

3.5.2 (Thm 27.3) A subspace A of \mathbb{R}^{n} is compact iff it is closed and is bounded in the Euclidean metric d or the square metric \rho.

3.5.3 (Extreme value theorem) (Thm 27.4) Let f: X -> Y be continuous, where Y is an ordered set in the order topology. If X is compact, then there exists points c and d in X s.t. f(c) \le f(x) \le f(d) for every x \in X.

3.5.4 Distance from x to A ( a point x \in X, nonempty subset A of X) : For a metric space (X,d), d(x,A) = inf {d(x,a) | a \in A}

3.5.4.1 D(x,A) is continuous

3.5.5 (Lebesgue number lemma)(Lem 27.5) Let \mathcal{A} be an open covering of the metric space (X,d). If X is compact, there is a \delta > 0 s.t. for each subset of X having diameter less than \delta, there exists an element of \mathcal{A} containing it.

3.5.5.1 Lebesgue number for the covering \mathcal{A} : \delta

3.5.6 Uniformly continuous (a function f : X -> Y) : Let metric spaces (X,d_{X}), (Y,d_{y}), if given \epsilon >0, there is a \delta >0 s.t. for every pair of points x_{0}, x_{1} of X, d_{X} (x_{0}, x_{1}) < \delta -> d_{Y} (f (x_{0}), f (x_{1}) ) < \epsilon

3.5.7 (Uniform continuity theorem) (Thm 27.6) : Let f : X -> Y be a continuous map of the compact metric space (X,d_{X}) to the metric space (Y,d_{Y}). Then f is uniformly continuous.

3.5.7.1 Pf) (Lebesgue number lemma)

3.5.8 Isolated point x of X (a point x of a space X) : one-point set {x} is open in X.

3.5.9 (Thm 27.7) Let X be a nonempty compact Hausdorff space. If X has no isolated points, then X is uncountable.

3.5.9.1 Pf) (Thm 26.9)

3.5.9.2 (Cor 27.8) Every closed interval in \mathbb{R} is uncountable.

3.6 Limit point compactness

3.6.1 Limit point compact (a space X) : every infinite subset of X has a limit point.

3.6.2 (Thm 28.1) Compactness implies limit point compactness, but not conversely.

3.6.3 Subsequence ( a sequence (x_{n}) of points of a topological space X) : n_{1} < n_{2} < … < n_{i} < … is an increasing sequence of positive integer, then the sequence (yi) defined by setting y_{i} = x_{n}

3.6.4 Sequentially compact ( a space X) : every sequence of points of X has a convergent subsequence

3.6.5 (Thm 28.2) Let X be a metrizable space. Then the following are equivalent.

3.6.5.1 X is compact.

3.6.5.2 X is limit point compact.

3.6.5.3 X is sequentially compact.

3.6.5.4 Pf) If X is sequentially compact, the (Lebesgue number lemma) holds for X.

3.7 Local compactness

3.7.1 Locally compact at x ( a point x in a space X) : there is some compact subspace C of X that contains a neighborhood of x.

3.7.2 Locally compact (a space X) : X is locally compact at each of its points

3.7.3 (Thm 29.1) Let X be a space. Then X is locally compact Hausdorff iff there exists a space Y satisfying the following conditions. If Y and Y’ are two spaces satisfying these conditions, then there is a homeomorphism of Y with Y’ that equals the identity map on X.

3.7.3.1 X is a subspace of Y.

3.7.3.2 The set Y-X consists of a single point.

3.7.3.3 Y is a compact Hausdorff space.

3.7.4 Compactification Y of X ( a compact Hausdorff space Y and a proper subspace X of Y) : closure of X equals Y

3.7.5 One-point compactification Y of X( a compact Hausdorff space Y and a proper subspace X of Y ): Y-X equals a single point

3.7.5.1 X has a one-point compactification Y iff X is a locally compact Hausdorff space that is not itself compact.

3.7.5.2 Riemann sphere : \mathbb{C} \cup {\infty} , one-point compactification of \mathbb{R}^{2}

3.7.6 (Thm 29.2) Let X be a Hausdorff space. Then X is locally compact iff given x in X, and given a neighborhood U of x, there is a neighborhood V of x s.t. \bar{V} is compact and \bar{V} \subset U.

3.7.6.1 (Cor 29.3) Let X be locally compact Hausdorff; let A be a subspace of X. If A is closed in X or open in X, then A is locally compact.

3.7.6.2 (Cor 29.4) A space X is homeomorphic to an open subspace of a compact Hausdorff space iff X is locally compact Hausdorff.

3.7.6.2.1 Pf) (Thm 29.1) (Cor 29.3)