Download MATH313 VUW - Topology and more Lecture notes Mathematics in PDF only on Docsity!
MATH 313 Notes
Max Christie
June 29, 2017
Topological Spaces Definition 1. Let X be a set. A topology T on X is a collection of subsets of X - which we call open sets - such that
(i) ∅ and X are open sets; (ii) The intersection of finitely many open sets is an open set; (iii) The union of any collection of open sets is an open set. Definition 1.1. A topological space is made up of a set X, and a topology T on X. Formal notation of a space is (X, T ) but often we shorten this just to saying X is a space, implying the topology T on X. Example 1. The trivial (or indiscrete) topology on X is just the collection of as few subsets of X as possible. Those are X and ∅. So the trivial topology would be T = {X, ∅} Example 1. The discrete topology on X is the collection of all subsets of X (i.e. the power set of X). T = P(X) = {U : U ⊆ X} Example 1. The finite complement topology T is the collection of all subsets of X that satisfy the following condition to be open: A ⊆ R is open in Xf c iff A = ∅ or (R − A) is finite.
Definition 1. If T 1 and T 2 are topologies on X, and T 1 ⊆ T 2 , then we call T 2 finer/stronger/larger than T 1 (or we call T 1 coarser/weaker/smaller than T 2 ). If T 2 is finer than T 1 and not equal to T 1 (T 1 ⊂ T 2 ) then it is strictly finer than T 1. If T 1 is coarser than T 2 and not equal to T 2 (T 1 ⊂ T 2 ) then it is strictly coarser than T 2.
Definition 1. A neighbourhood (nhd) of a point x in a topological space is any open set U that contains x. In other words: an nhd of x is any U ⊆ T with x ∈ U.
Theorem 1. In any topological space X, a subset A of X is open iff every member of A has a neighbourhood within A. In other words, a subset of X is open iff it is made up of a union of open sets inside it.
Definition 1. If X is any set, a collection B of subsets of X is a basis of X if the following hold:
(i) For every x ∈ X, there is a B ∈ B such that x ∈ B. (ii) If B 1 and B 2 are in B and x ∈ B 1 ∩ B 2 then there is a B 3 ∈ B such that x ∈ B 3 ⊂ B 1 ∩ B 2. Definition 1. The topology TB on X generated by the basis B is defined by: for any U ⊆ X, U ∈ TB iff (i) U = ∅ or (ii) U is a union of a collection of members of B
Theorem 1. If B is a basis on set X, then TB is a topology, where U ∈ TB iff ∀x ∈ U ∃ Bx ∈ B with x ∈ Bx ⊆ U. Example 1. If the basis is the collection of half open intervals with a closed left bound and open right bound: B = {[a, b) ⊂ R|a < b} then the topology generated by it is the lower limite topology, where all open sets are unions of the basis sets.
Theorem 1. If B is a basis for a topology T on set X, then U is open in T iff for every x ∈ U there is a basis element such that x ∈ B ⊆ U.
Theorem 1. If for every x ∈ U ⊂ X there is an open set V ∈ C such that x ∈ V ⊆ U , then C is a basis that generates the topology.
Definition 1. A subset A of a topological space X is closed if the set X − A is open.
Theorem 1. Let X be a topological space. The following statements hold:
(i) ∅ and X are closed.
(ii) The intersection of any collection of closed sets is a closed set. (iii) The union of finitely many closed sets is a closed set. Definition 1. A topological space X is Hausdorff iff for every pair of distinct points x and y in X there exist disjoint neighbourhoods U and V such that x ∈ U and y ∈ V.
Theorem 1. If X is a Hausdorff space, then every single-point subset of X is closed.
Interior, Closure, and Boundary
Definition 2. Let A ⊆ X. The interior of A is the union of all open sets contained in A. The closure of A is the intersection of all closed sets containing A.
Theorem 2. Let X be a topological space and A and B be subsets of X.
(i) δA is closed.
(ii) δA = Cl(A) ∩ Cl(X − A)
(iii) δA ∩ Int(A) = ∅
(iv) δA ∪ Int(A) = Cl(A)
(v) δA ⊂ A iff A is closed
(vi) δA ∩ A = ∅ iff A is open
(vii) δA = ∅ iff A is both open and closed
Creating New Topological Space
Definition 3. Let Y ⊆ X. Define TY = {U ∩ Y |U ∈ TX } as the subspace topology on Y , making (Y, TY ) a subspace of X.
Theorem 3. Let Y ⊆ X. Then X ⊆ Y is closed in Y iff C = D ∩ Y for some closed set in X.
Theorem 3. Let X be a topological space and B be a basis for TX. If Y ⊆ X, then BY = {B ∩ Y |B ∈ B} is a basis for TY.
Definition 3. If X × Y is the product of topological spaces then the product topology is generated by the basis B = {U × V | U ∈ TX and V ∈ TY }
Theorem 3. If C is a basis for X and D is a basis for Y , then E = {C × D|C ∈ C and D ∈ D} is a basis that generates the product topology on X × Y.
Theorem 3. Let A and B be subsets of topological spaces X and Y respectively. Then Int(A × B) = Int(A) × Int(B).
Quotient Topology
Definition 3. Let p : X → A be a surjective map. Define U ⊂ A to be open in A iff p−^1 (U ) is open in X. The collection of open sets in A is the quotient topology induced by p.
Definition 0.16 Let R be an equivalence relation on a set X. Then R ⊆ X × X = {< x, y >: x, y ∈ X}. Can write xRy when < x, y >∈ R. R has the following properties:
- reflexive: xRx ∀x ∈ X
- symmetric: xRy implies yRx
- transitive: xRy and yRz implies xRz
Definition 0. The equivalence class of an x ∈ X is [x]R = {y ∈ C : xRy}. Facts:
- x ∈ [x]R
- [x]R = [y]R iff xRy
- If [x]R 6 = [y]R then [x]R ∩ [y]R = ∅ (i.e. distinct equivalence classes are disjoint)
Definition 0. A partition of X is a collection P of subsets of X that are (i) non-empty, (ii) mutually disjoint, and (iii) whose union is X. Members of P may be called cells. (Mutually disjoint: if C, C′^ ∈ P and C 6 = C′^ then C ∩ C′^ = ∅). If R is an equivalence relation on X, the set X − R = {[x]R : x ∈ X} of all R-equivalence classes is a partition of X, the quotient set of X by R. Conversely, given a partition P ⊆ P(x), define RP ⊆ X × X by: < x, y >∈ RP iff x and y belong to the same cell of P. Then RP is an equivalence relationship on X for which [x]RP is the cell containing x. This gives a bijective correspondence between the set of all equivalence relations on X and the set of all partitions of X. The quotient map fR : X → X/R is defined by fR(x) = [x]R. fR is surjective (onto X/R). This means that for any C ∈ X/R, ∃x ∈ X such that C = [x]R = fR. Similarly, given a partition P we have a surjective function fP : X → P with fP is the cell containing x. Define Rf ⊆ X ×X by xRf z iff f (x) = f (z) ∀x, z ∈ X. Then Rf is an equivalence relation on X and [x]Rf = {z ∈ X : f (x) = f (z)} ⊆ X. Given any y ∈ Y , ∃x ∈ X s.t. f (x) = y as f is surjective, then [x]Rf = {z ∈ X : f (z) = y} = f −^1 (y). For W ⊆ Y , define f −^1 (W ) = {x ∈ X : f (x) ∈ W } as the preimage of W under f. Define g : X/Rf → Y s.t. g([x]Rf ) = f (x) If [x]Rf = [z]Rf , then xRf z, so f (x) = f (z)
- g is injective: if g([x]Rf ) = g([z]Rf ) then f (x) = f (z), soxRf z, so [x]Rf = [z]Rf
- g is surjective: take y ∈ Y. Then y = f (x) for some x ∈ X as f is surjective. So y = f (x) = g([x]Rf )
Definition 0. Given f : X → Y and a subset U of Y whe have f −^1 (U ) = {x ∈ X : f (x ∈ B)} In other words, {x ∈ X : ∃y ∈ Y (y = f (x))} Which is the preimage of U under f Have f (U )−^1 = ∪{f −^1 y : y ∈ U where f −^1 = {x ∈ X : f (x) = y} Case of partiton P of X Have fP : X → P with fP (x) = C ∈ P iff x ∈ C If C ∈ P then C ⊆ X is the point preimage f (^) P− 1 (C) = {x ∈ X : fP (X) = C}
Defintion 4. A function f : X → Y is an embedding if it makes X homeomorphic to the subspace f (x) of Y.
Theorem 4. If f : X → Y is a homeomorphism and X is Hausdorff, then Y is Hausdorff.
Metrics
Definition 5. A metric on a set X is a function d : X × X → R with the following properties:
(i) d(x, y) ≥ 0 for all x, y ∈ X
(ii) d(x, y) = d(y, x) for all x, y ∈ X (iii) d(x, y) + d(y, z) ≥ d(x, z) for all x, y, z ∈ X Example 5.1, 5.2, 5. The standard metric on R is defined as d(x, y) = |x − y|. The standard metric on R^2 is defined as d(p, q) =
(p 1 − q 1 )^2 + (p 2 − q 2 )^2. The taxicab metric on R^2 is defined as dT (p, q) = |p 1 − q 1 | + |p 2 − q 2 |. The max metric on R^2 is defined as dM (p, q) = max{|p 1 − q 1 |, |p 2 − q 2 |}.
Theorem 5. Let (X, d) be a metric space. The collection of open balls, B = {Bd(x, )|x ∈ X, > 0 }, is a basis for a topology on X.
Lemma 5. Let (X, d) be a metric space. If x ∈ X, > 0, and y ∈ Bd(x, ) then there exists δ > 0 such that Bd(y, δ) ⊂ Bd(x, )
Definition 5. Let (X, d) be a metric space. The topology generated by the basis of open balls B = {Bd(x, )|x ∈ X, > 0 } is called the topology induced by d and is referred to as the matrix topolgy.
Theorem 5. Let (X, d) be a metric space. A set U ⊂ X is open in the topology induced by D iff for every y ∈ U there is a δ > 0 such that Bd(y, δ) ⊂ U.
Definition 5. Let (X, dX ) and (Y, dY ) be metric spaces. A bijective function f : X → F is called an isometry if dX (x, x′) = dy(f (x), f (x′)). (Preserves distance). If two metric spaces (X, dX ) and (Y, dY ) are isometric then the equivalent topological spaces (X, TX ) and (Y, TY ) are homeomorphic. Isometry is a stricter notion than homeomorphism: so two homeomorphic spaces are not necessarily metric spaces. Theorem 5. Every metric space is Hausdorff. Definition 5. Let X be a topological space. We say X is metrizable if there exists a metric d on X that induces the topology on X.
Theorem 5. If X is a metrizable topological space and Y is homeomorphic to X, then Y is metrizable.
Connectedness
Definition 6. Let X be a topological space.
(i) We call X connected if there does not exist a pair of disjoint nonempty open sets whose union is X.
(ii) We call X disconnected if X is not connected. (iii) If X is disconnected, then there is a pair of disjoint nonempty open sets whose union is X called a separation of X.
Looking at this we can also say that X is connected if there is no separation of X, and X is dis- connected if there is a separation of X.
Theorem 6. A topological space X is connected iff there are no nonempty proper subsets of X (not equal to X) that are both open and closed in X.
Definitino X Subsets A and B of space X are separated from each other if A ∩ ClX (B) = ∅ and B ∩ ClX (A) = ∅.
Theorem X X is connected iff it is not the union of two separated subsets.
Definition 6. A set A ⊆ X is connected in X if A is connected in the subspace topology.
Theorem 6. A set A is disconnected in X iff there are open subsets U, V ⊆ X such that A ⊂ U ∪ V , and U ∩ A 6 = ∅, and V ∩ A 6 = ∅, and U ∩ V ∩ A = ∅.
Definition 6. These open sets U and V from Th 6.4 are called a separation of A in X.
Theorem 6. If X is connected and f : X → Y is continuous, then f (X) is connected in Y.
Definition X A subset A ⊆ R is convex if, for all a, b ∈ A with a ≤ b, the interval (a, b) ⊆ A.
Theorem X A subset A ⊆ R is connected in the standard topology iff it is convex.
Lemma 6. Let C ⊆ D ⊆ X, with C a connected set, assuming U and V form a separation of D in X. Then
Compactness
Definition 7. Let A be a subset of space X, and O be a collection of subsets of X. Then:
(ii) O covers A if A ⊆ ∪O.
(ii) If O covers A, and each set in O is open, then O is an open cover of A.
(iii) If O covers A, and O′^ is a subcollection of A that also covers A, then O′^ is a subcover of O.
Definition 7. A topological space X is compact if every open cover of X has a finite subcover.
Defintion 7. Let X be a topological space, and A ⊆ X. Then A is compact in X if A is compact in the subspace topology inherited from X.
Lemma 7. Assume A ⊆ X. Then A is compact in X iff every cover of A by X-open sets has a finite subcover.
Theorem 7. Let f : X → Y be continuous, and let A be compact in X. Then f (A) is compact in Y.
Theorem 7. If D is a compact subset of space X then a closed subset C of D is compact.
Theorem 7. If X is a Hausdorff space and A is is compact in X. Then A is closed in X.
Lemma 7. Let X and Y be topological spaces and Y be compact. If x ∈ X, and U is an open set in X × Y containing {x}timesY }n then there exists a neighbourhood W of x in X such that W × Y ⊆ U.
Theorem 7. If X and Y are compact topological spaces, then the product X × Y is compact.
Corollary 7. Let X 1 , · · · , Xn be topological spaces, and let Ai be a compact subset of Xi for each i = 1, · · · , n. Then A 1 × · · · × An is a compact subset of the product space X 1 × · · · × Xn.
Lemma 7. Let {[an, bn]}n∈Z+ be a collection of non-empty closed bounded intervals in R such that [an+1, bn+1] ⊆ [an, bn] for each n ∈ Z+. Then ∩∞ n=1[an, bn] is non-empty.
Theorem 7. Every closed and bounded interval [a, b] is a compact subset of R with the standard topology.
Definition Y In a metric space (X, d), a subset A of X is bounded if there exists a real number m > 0 such that d(x, y) < m for all x, y ∈ A. Then given some a ∈ A, A ⊆ Bd(a, m). Heine-Borel Theorem A subset A of Rnstd is compact iff it is closed and bounded.
Theorem 7. Let X be compact and f : X → R be continuous. Then f takes on a maximum value and a minimum value on X (there exist a, b ∈ X such that f (a) ≤ f (x) ≤ f (b) for all x ∈ X).
Missed lecture 23/
Definition 9. Let f, g : X → Y be continuous functions. Assume that I = [0, 1] has the subspace topology it inherits from R and that X × I has the product topology. A homotopy from f to g is a continuous function F : X × I → Y such that F (x, 0) = f (x) and F (x, 1) = g(x). Notated f ' g
Theorem 9. ' is an equivalence relation on the set of all continuous functions f : X → Y.
Definition 9. Let C(X, Y ) denote the set of all continuous functions f : X → Y. The homotopy classes in C(X, Y ) are the equivalence classes under the relation ', denoted [f ].
Straight Line Theorem If A is convex, then for any space X, any two continuous functions f, g : X → A are homotopic by the function F : X × I → A where F (x, t) = f (x) + t(g(x) − f (x)). F is called a straight-line homotopy
Definition 9. A space X is contractible if the identity function idX : X → X is null-homotopic. A function is null-homotopic if it is homotopic to a constant function kp : X → X for some p ∈ X, where kp(x) = p for all x ∈ X.
Definition of Star Convex A subset A of Rn^ is star convex if there exists a point p∗ ∈ A such that for every p ∈ A, the line segment in Rn^ joinig p∗ and p lies in A.
Defintion of Path Homotopy A path homotopy from f to g (loops that both have f (0) = f (1) = g(0) = g(1) = p) defines a continuous function F : I × I → X such that F (x, 0) = f (x), F (x, 1) = g(x), F (0, t) = p = F (1, t). Then [f ]p is the path homotopy class of loop f among all loops based at p.
Definition of an Operation The operation ∗ on Π 1 (X, p) is [f ]p ∗ [g]p = [f ∗ g]p where f ∗ g is the concatenation of f and g. This operation is well defined and associative.
Theorem
