Appendix A Topological Spaces and Manifolds A.1 Topology and Metric A subset of R is called open if it is the union of open intervals (a, b) ⊆ R.This definition makes R into a topological space: A.1 Definition 1. A topological space is a pair (M, O), consisting of a set M and a set O of subsets of M (these subsets will be called open sets) such that the following properties are satisfied: 1. ∅ and M are open, 2. arbitrary unions of open sets are open, 3. the intersection of any two open sets is open. 2. O is called a topology on M. 3. If O1 and O2 are topologies on M and O1 ⊆ O2, then we call O1 coarser or weaker than O2, and O2 finer or stronger than O1. A.2 Example 1. The discrete topology 2M (the power set) is the finest topology onasetM, and the trivial topology {M, ∅} is the coarsest topology on M. Topological spaces (M, 2M ) are called discrete. 2. If N ⊆ M is a subset of the topological space (M, O), then {U ∩ N | U ∈ O}⊆2N (A.1.1) defines a topology on N, called the subspace topology, relative topology, induced topology,ortrace topology. For example, for the subset N := [0, ∞) of R,all subintervals of the form [0, b) ⊂ N with b ∈ N are open in N, but not open in R. 3. If (M, OM ) and (N, ON ) are disjoint topological spaces (i.e., M ∩ N =∅), then M∪N the union M ∪ N with {U ∪ V | U ∈ OM , V ∈ ON }⊆2 is a topological space, called the sum space. © Springer-Verlag GmbH Germany 2018 483 A. Knauf, Mathematical Physics: Classical Mechanics, La Matematica per il 3+2 109, https://doi.org/10.1007/978-3-662-55774-7 484 Appendix A: Topological Spaces and Manifolds 4. Let (M, OM ) be a topological space and f : M → N surjective. The quotient topology induced by f on N is the topology −1 N V ⊆ N | f (V ) ∈ OM ⊆ 2 . ♦ A.3 Theorem Let F be an arbitrary family of subsets of a set M. • Then there is a unique coarsest topology O(F) on M satisfying F ⊆ O(F). • O(F) is then called the topology generated by F. In many cases, the topology is generated by a metric. A.4 Definition • A metric space is a pair (M, d), consisting of a set M and a function d : M × M → [0, ∞), such that the following properties hold for all x, y, z ∈ M: (a) d(x, y) = 0 ⇐⇒ x = y (Positivity) (b) d(x, y) = d(y, x) (Symmetry) (c) d(x, z) ≤ d(x, y) + d(y, z) (Triangle Inequality). • d is called a metric,d(x, y) the distance between x and y. • For x ∈ M and ε > 0,theset Uε(x) := {y ∈ M | d(y, x)<ε} is called the (open) ε-neighborhood of xinM. A.5 Example 1. The set B := {0, 1} designates one bit. We consider sequences of n ∈ N bits, i.e., elements1 of Bn. Their Hamming distance is given by the metric d : Bn × Bn →{0, 1,...,n}, d (b1,...,bn), (c1,...,cn) := i ∈{1,...,n}|bi = ci , namely by the number of positions in which the bit sequences differ. This metric is used in information theory. 2. If (M, d) is a metric space and N a subset of M, then (N, dN ) is again a metric space with the metric ‘restricted’2 to N, dN : N × N → R , dN (x, y) := d(x, y). For instance, for M := C with the metric d(x, y) := |x − y|,thesetS1 ⊂ C is geometrically speaking the circle of radius 1 about the origin. The distance dS1 (x, y) between two points on the circle is then the length of the straight line segment between the points x and y. A different, sensible metric on S1 is given as the (minimal) difference in angles of x and y. 1We will write them as row vectors. 2More precisely, it is the mapping d : M × M → R that is restricted to N × N. Appendix A: Topological Spaces and Manifolds 485 3. On the vector space Rn, the length of a vector v is defined by the Euclidean norm v:= n v2 v = (v ,...,v ) ∈ Rn . i=1 i 1 n With d(x, y) := y − x, one obtains from it a metric on Rn, called the Euclid- ean metric. 4. On the vector space Rn, one can also define a metric in terms of the maximum norm n v∞ := max(|v1|,...,|vn|) v = (v1,...,vn) ∈ R . d∞(x, y) := x − y∞, and d∞ satisfies √ n d∞(x, y) ≤ d(x, y) ≤ nd∞(x, y)(x, y ∈ R ); see the figure for n = 2 and x = 0. n The ‘unit ball’ {v ∈ R |v∞ ≤ 1} with respect to the maximum norm is an n-dimensional cube of side length 2 that is parallel to the axes and centered in the origin. ♦ A.6 Definition For a metric space (M, d),theset O(d) := V ⊆ M |∀x ∈ V ∃ε > 0 : Uε(x) ⊆ V . (A.1.2) is called the (metric) topology on (M, d). A.7 Theorem M, O(d) is a topological space, and the ε-neighborhoods Uε(x) are open. A.8 Remark (Topological Vector Spaces) Frequently, different metrics generate the same topology. This is in particular true for such metrics on vector spaces as arise from equivalent norms3 (as in Examples A.5.3 and A.5.4). For in this case, one can find for every ε-ball around x with respect to one norm an ε/c-ball around x with respect to the other norm, that is contained in the former ball. As all norms on a finite dimensional vector space are equivalent, we can talk about the topology of Rn or Cn (however, there are other topologies on these spaces, but they do not arise from norms, see Example A.2.1). On an infinite dimensional K-vector space V , there are many different topologies O that are generated by norms. In any case, addition and multiplication by scalars are continuous operations, so (V, O) is called a topological vector space. ♦ 3 Definition: Two norms ·I , ·II : V → R are called equivalent if there exists a number c ≥ 1 with −1 c vI ≤vII ≤ cvI (v ∈ V ). 486 Appendix A: Topological Spaces and Manifolds We expand our topological vocabulary by generalizing notions that are familiar from the space R: A.9 Definition Let (M, O) be a topological space. • A ⊆ M is called closed if A is the complement of an open set: M \ A ∈ O. • U ⊆ M is called a neighborhood of x ∈ M if there exists an open set V with x ∈ V ⊆ U. • For A ⊆ M and x ∈ M, we call x an interior resp. exterior resp. boundary point of A, depending on whether A or M \ A or neither of the two sets is a neighborhood of x. – A˚ := {x ∈ M | x is interior point of A} is called the interior of A. – A := {x ∈ M | x not exterior point of A} is called the closure or the closed hull of A. – ∂ A := {x ∈ M | x boundary point of A} is called the boundary of A. • x ∈ M is called cluster point of the subset A ⊆ MifthesetU∩ (A \{x}) is not empty for any neighborhood U of x. • N ⊆ M is called dense if N = M. • N ⊆ M is called nowhere dense if the interior of Nisempty. A.10 Example 1. For A := (0, 1]⊆R,wehaveA˚ = (0, 1), A =[0, 1], and ∂ A ={0, 1}. 2. Q is dense in R, and Z is nowhere dense in R. ♦ A.11 Definition Let (M, O) be a topological space. • A family (Ui )i∈I of Ui ∈ O is called an open cover of M if = . i∈I Ui M • (M, O) is called compact if every open cover (Ui )i∈I has a finite subcover, i.e., an open cover (U j ) j∈J with a finite index set J ⊆ I. • (M, O) is called locally compact if all m ∈ M have a compact neighborhood. • (M, O) is called a Hausdorff space if for all x = y ∈ M, there exist disjoint neighborhoods Ux of x and Uy of y. • (M, O) is called paracompact if it is Hausdorff and for every open cover {Ui }i∈I , there exists an open cover {Vj } j∈J that is (a) a refinement of {Ui }i∈I , i.e., ∀ j ∈ J ∃i ∈ I : Vj ⊆ Ui , (b) and that is locally finite, i.e., for all x ∈ M there exists a neighborhood U of x for which the index set { j ∈ J | Vj ∩ U =∅}is finite. A.12 Example (Topological Notions) 1. Finite dimensional real and complex vector spaces V are locally compact and paracompact. Subsets of these V are compact if and only if they are closed and bounded. (The latter is called the Heine-Borel theorem.) However, Q with the relative topology from R is not locally compact. Appendix A: Topological Spaces and Manifolds 487 2. As in a metric space (M, d), all points x = y ∈ M have a positive distance, the topological space M, O(d) is Hausdorff. A topological space (M, O) is called metrizable if there exists a metric d on M for which O = O(d). The example of spaces that are not Hausdorff shows that not every topological space is metrizable. 3. Infinite-dimensional Banach spaces are not locally compact, but are paracompact (as is every metric space).