Lecture 05 - Topological Spaces: Some Heavily Used Invariants (Schuller's Geometric Anatomy of Theoretical Physics)

topological spaces: some heavily used invariants

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topological spaces: some heavily used invariants

5.1 Separation properties Definition. A topological space (M, O) is said to be T1 if for any two distinct points p, q ∈ M, p 6= q: ∃U ∈ O : p ∈ U∧q ∈ / U. Definition. A topological space (M, O) is said to be T2 or Hausdorff if, for any two distinct points, there exist non-intersecting open neighbourhoods of these two points: ∀ p, q ∈ M : p 6= q ⇒ ∃ U(p), V(q) ∈ O : U(p) ∩ V(q) = ∅. Example 5.1. The topological space (Rd , Ostd ) is T2 and hence also T1. Example 5.2. The Zariski topology on an algebraic variety is T1 but not T2. Example 5.3. The topological space (M, {∅, M}) does not have the T1 property since for any p ∈ M, the only open neighbourhood of p is M and for any other q 6= p we have q ∈ M. Moreover, since this space is not T1, it cannot be T2 either. Remark 5.4. There are many other “T” properties, including a T21⁄2 property which differs from T2 in that the neighbourhoods are closed. 5.2 Compactness and paracompactness Definition. Let (M, O) be a topological space. A set C ⊆ P(M) is called a cover (of M) if: [ C = M. Additionally, it is said to an open cover if C ⊆ O. e ⊆ C such that C e is still a cover, is Definition. Let C be a cover. Then any subset C called a subcover. Additionally, it is said to be a finite subcover if it is finite as a set. Definition. A topological space (M, O) is said to be compact if every open cover has a finite subcover. Definition. Let (M, O) be a topological space. A subset N ⊆ M is called compact if the topological space (N, O|N ) is compact. Determining whether a set is compact or not is not an easy task. Fortunately though, for Rd equipped with the standard topology Ostd , the following theorem greatly simplifies matters. Theorem 5.5 (Heine-Borel). Let Rd be equipped with the standard topology Ostd . Then, a subset of Rd is compact if, and only if, it is closed and bounded.

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topological spaces: some heavily used invariants A subset S of Rd is said to be bounded if: ∃ r ∈ R+ : S ⊆ Br (0). Remark 5.6. It is also possible to generalize this result to arbitrary metric spaces. A metric space is a pair (M, d) where M is a set and d : M × M → R is a map such that for any x, y, z ∈ M the following conditions hold: i) d(x, y) > 0; ii) d(x, y) = 0 ⇔ x = y; iii) d(x, y) = d(y, x); iv) d(x, y) 6 d(x, z) + d(y, z). A metric structure on a set M induces a topology Od on M by: U ∈ Od :⇔ ∀ p ∈ U : ∃ r ∈ R+ : Br (p) ⊆ U, where the open ball in a metric space is defined as: Br (p) := {x ∈ M | d(p, x) < r}. In this setting, one can prove that a subset S ⊆ M of a metric space (M, d) is compact if, and only if, it is complete and totally bounded. Example 5.7. The interval [0, 1] is compact in (R, Ostd ). The one-element set containing (−1, 2) is a cover of [0, 1], but it is also a finite subcover and hence [0, 1] is compact from the definition. Alternatively, [0, 1] is clearly closed and bounded, and hence it is compact by the Heine-Borel theorem. Example 5.8. The set R is not compact in (R, Ostd ). To prove this, it suffices to show that there exists a cover of R that does not have a finite subcover. To this end, let: C := {(n, n + 1) | n ∈ Z} ∪ {(n + 12 , n + 32 ) | n ∈ Z}. This corresponds to the following picture.

C −1

−1/2

0

1/2

1

R

It is clear that removing even one element from C will cause C to fail to be an open cover of R. Therefore, there is no finite subcover of C and hence, R is not compact.

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topological spaces: some heavily used invariants Theorem 5.9. Let (M, OM ) and (N, ON ) be compact topological spaces. Then (M × N, OM×N ) is a compact topological space. The above theorem easily extends to finite cartesian products. Definition. Let (M, O) be a topological space and let C be a cover. A refinement of C is a cover R such that: ∀ U ∈ R : ∃ V ∈ C : U ⊆ V. Any subcover of a cover is a refinement of that cover, but the converse is not true in general. A refinement R is said to be: • open if R ⊆ O; • locally finite if for any p ∈ M there exists a neighbourhood U(p) such that the set: {U ∈ R | U ∩ U(p) 6= ∅} is finite as a set. Compactness is a very strong property. Hence often times it does not hold, but a weaker and still useful property, called paracompactness, may still hold. Definition. A topological space (M, O) is said to be paracompact if every open cover has an open refinement that is locally finite. Corollary 5.10. If a topological space is compact, then it is also paracompact. Definition. A topological space (M, O) is said to be metrisable if there exists a metric d such that the topology induced by d is precisely O, i.e. Od = O. Theorem 5.11 (Stone). Every metrisable space is paracompact. Example 5.12. The space (Rd , Ostd ) is metrisable since Ostd = Od where d = k · k2 . Hence it is paracompact by Stone’s theorem. Remark 5.13. Paracompactness is, informally, a rather natural property since every example of a non-paracompact space looks artificial. One such example is the long line (or Alexandroff line). To construct it, we first observe that we could “build” R by taking the interval [0, 1) and stacking countably many copies of it one after the other. Hence, in a sense, R is equivalent to Z × [0, 1). The long line L is defined analogously as L : ω1 × [0, 1), where ω1 is an uncountably infinite set. The resulting space L is not paracompact. Theorem 5.14. Let (M, OM ) be a paracompact space and let (N, ON ) be a compact space. Then M × N (equipped with the product topology) is paracompact. Corollary 5.15. Let (M, OM ) be a paracompact space and let (Ni , ONi ) be compact spaces for every 1 6 i 6 n. Then M × N1 × . . . × Nn is paracompact.

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topological spaces: some heavily used invariants Definition. Let (M, OM ) be a topological space. A partition of unity of M is a set F of continuous maps from M to the interval [0, 1] such that for each p ∈ M the following conditions hold: i) there exists U(p) such that the set {f ∈ F | ∀ x ∈ U(p) : f(x) 6= 0} is finite; P ii) f∈F f(p) = 1. If C is an open cover, then F is said to be subordinate to the cover C if: ∀ f ∈ F : ∃ U ∈ C : f(x) 6= 0 ⇒ x ∈ U. Theorem 5.16. Let (M, OM ) be a Hausdorff topological space. Then (M, OM ) is paracompact if, and only if, every open cover admits a partition of unity subordinate to that cover. Example 5.17. Let R be equipped with the standard topology. Then R is paracompact by Stone’s theorem. Hence, every open cover of R admits a partition of unity subordinate to that cover. As a simple example, consider F = {f, g}, where:   0 if x 6 0 f(x) = x2 if 0 6 x 6 1  1 if x > 1

 if x 6 0  1 2 and g(x) = 1 − x if 0 6 x 6 1  0 if x > 1

Then F is a partition of unity of R. Indeed, f, g : R → [0, 1] are both continuous, condition i) is satisfied since F itself is finite, and we have ∀ x ∈ R : f(x) + g(x) = 1. Let C := {(−∞, 1), (0, ∞)}. Then C is an open cover of R and since: f(x) 6= 0 ⇒ x ∈ (0, ∞)

and

g(x) 6= 0 ⇒ x ∈ (−∞, 1),

the partition of unity F is subordinate to the open cover C. 5.3 Connectedness and path-connectedness Definition. A topological space (M, O) is said to be connected unless there exist two non-empty, non-intersecting open sets A and B such that M = A ∪ B. Example 5.18. Consider (R \ {0}, Ostd |R\{0} ), i.e. R \ {0} equipped with the subset topology inherited from R. This topological space is not connected since (−∞, 0) and (0, ∞) are open, non-empty, non-intersecting sets such that R \ {0} = (−∞, 0) ∪ (0, ∞). Theorem 5.19. The interval [0, 1] ⊆ R equipped with the subset topology is connected. Theorem 5.20. A topological space (M, O) is connected if, and only if, the only subsets that are both open and closed are ∅ and M.

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topological spaces: some heavily used invariants Proof. (⇒) Suppose, for the sake of contradiction, that there exists U ⊆ M such that U is both open and closed and U ∈ / {∅, M}. Consider the sets U and M \ U. Clearly, we have U ∩ M \ U = ∅. Moreover, M \ U is open since U is closed. Therefore, U and M \ U are two open, non-empty, non-intersecting sets such that M = U ∪ M \ U, contradicting the connectedness of (M, O). (⇐) Suppose that (M, O) is not connected. Then there exist open, non-empty, nonintersecting subsets A, B ⊆ M such that M = A ∪ B. Clearly, A 6= M, otherwise we would have B = ∅. Moreover, since B is open, A = M \ B is closed. Hence, A is a set which is both open and closed and a ∈ / {∅, M}. Definition. A topological space (M, O) is said to be path-connected if for every pair of points p, q ∈ M there exists a continuous curve γ : [0, 1] → M such that γ(0) = p and γ(1) = q. Example 5.21. The space (Rd , Ostd ) is path-connected. Indeed, let p, q ∈ Rd and let: γ(λ) := p + λ(q − p). Then γ is continuous and satisfies γ(0) = p and γ(1) = q. Example 5.22. Let S := {(x, sin( x1 )) | x ∈ (0, 1]} ∪ {(0, 0)} be equipped with the subset topology inherited from R2 . Then (S, Ostd |S ) is connected but not path-connected. Theorem 5.23. If a topological space is path-connected, then it is also connected. Proof. Let (M, O) be path-connected but not connected. Then there exist open, nonempty, non-intersecting subsets A, B ⊆ M such that M = A ∪ B. Let p ∈ A and q ∈ B. Since (M, O) is path-connected, there exists a continuous curve γ : [0, 1] → M such that γ(0) = p and γ(1) = q. Then: [0, 1] = preimγ (M) = preimγ (A ∪ B) = preimγ (A) ∪ preimγ (B). The sets preimγ (A) and preimγ (B) are both open, non-empty and non-intersecting, contradicting the fact that [0, 1] is connected. 5.4 Homotopic curves and the fundamental group Definition. Let (M, O) be a topological space. Two curves γ, δ : [0, 1] → M such that: γ(0) = δ(0)

and γ(1) = δ(1)

are said to be homotopic if there exists a continuous map h : [0, 1] × [0, 1] → M such that for all λ ∈ [0, 1]: h(0, λ) = γ(λ) and h(1, λ) = δ(λ).

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topological spaces: some heavily used invariants

γ

q

h

p

δ Pictorially, two curves are homotopic if they can be continuously deformed into one another. Proposition 5.24. Let γ ∼ δ :⇔ γ and δ are homotopic. Then ∼ is an equivalence relation. Definition. Let (M, O) be a topological space. Then, for every p ∈ M, we define the space of loops at p by: Lp := {γ : [0, 1] → M | γ is continuous and γ(0) = γ(1)}. Definition. Let Lp be the space of loops at p ∈ M. We define the concatenation operation ∗ : Lp × Lp → Lp by: (γ ∗ δ)(λ) :=

γ(2λ) if 0 6 λ 6 12 δ(2λ − 1) if 12 6 λ 6 1

Definition. Let (M, O) be a topological space. The fundamental group π1 (p) of (M, O) at p ∈ M is the set: π1 (p) := Lp / ∼ = {[γ] | γ ∈ Lp }, where ∼ is the homotopy equivalence relation, together with the map • : π1 (p) × π1 (p) → π1 (p) defined by: [γ] • [δ] := [γ ∗ δ]. Remark 5.25. Recall that a group is a pair (G, •) where G is a set and • : G × G → G is a map (also called binary operation) such that: i) ∀ a, b, c ∈ G : (a • b) • c = a • (b • c); ii) ∃ e ∈ G : ∀ g ∈ G : g • e = e • g = g; iii) ∀ g ∈ G : ∃ g−1 ∈ G : g • g−1 = g−1 • g = e.

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topological spaces: some heavily used invariants A group isomorphism between two groups (G, •) and (H, ◦) is a bijection φ : G → H such that: ∀ a, b ∈ G : φ(a • b) = φ(a) ◦ φ(b). If there exists a group isomorphism between (G, •) and (H, ◦), we say that G and H ∼ grp H. are (group theoretic) isomorphic and we write G = The operation • is associative (since concatenation is associative); the neutral element of the fundamental group (π1 (p), •) is (the equivalence class of) the constant curve γe defined by: γe : [0, 1] → M λ 7→ γe (0) = p Finally, for each [γ] ∈ π1 (p), the inverse under • is the element [−γ], where −γ is defined by: −γ : [0, 1] → M λ 7→ γ(1 − λ) All the previously discussed topological properties are “boolean-valued”, i.e. a topological space is either Hausdorff or not Hausdorff, either connected or not connected, and so on. The fundamental group is a “group-valued” property, i.e. the value of the property is not “either yes or no”, but a group. A property of a topological space is called an invariant if any two homeomorphic spaces share the property. A classification of topological spaces would be a list of topological invariants such that any two spaces which share these invariants are homeomorphic. As of now, no such list is known. Example 5.26. The 2-sphere is defined as the set: S2 := {(x, y, z) ∈ R3 | x2 + y2 + z2 = 1} equipped with the subset topology inherited from R3 . The sphere has the property that all the loops at any point are homotopic, hence the fundamental group (at every point) of the sphere is the trivial group: ∀ p ∈ S2 : π1 (p) = 1 := {[γe ]}. Example 5.27. The cylinder is defined as C := R × S1 equipped with the product topology. A loop in C can either go around the cylinder (i.e. around its central axis) or not. If it does not, then it can be continuously deformed to a point (the identity loop). If it does, then it cannot be deformed to the identity loop (intuitively because the cylinder is infinitely long) and hence it is a homotopically different loop. The number of times a loop winds around the cylinder is called the winding number. Loops with different

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topological spaces: some heavily used invariants winding numbers are not homotopic. Moreover, loops with different orientations are also not homotopic and hence we have: ∼ grp (Z, +). ∀ p ∈ C : (π1 (p), •) = Example 5.28. The 2-torus is defined as the set T 2 := S1 × S1 equipped with the product topology. A loop in T 2 can intuitively wind around the cylinder-like part of the torus as well as around the hole of the torus. That is, there are two independent winding numbers and hence: ∼ grp Z × Z, ∀ p ∈ T 2 : π1 (p) = where Z × Z is understood as a group under pairwise addition.

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