Topological spaces and continuous functions¶
2 Topological spaces and continuous functions
2.1 Topological spaces
2.1.1 Topology on a set X : a collection \Tau of subsets of X having following properties.
2.1.1.1 \empty and X are in \Tau.
2.1.1.2 Union of the elements of any subcollection of \Tau is in \Tau.
2.1.1.3 Intersection of the elements of any finite subcollection of \Tau is in \Tau.
2.1.2 Topological space : ordered pair (X, \Tau)
2.1.2.1 X <- ordered pair (X, \Tau)
2.1.3 Open set of X (a subset U of X) : Let X is a topological space with topology \Tau, if U belongs to the collection \Tau.
2.1.4 Sort
2.1.4.1 Discrete topology ( a set X) : a collection of all subsets of X
2.1.4.2 Indiscrete topology ( a set X) : {X, \empty} ~ trivial topology
2.1.4.3 Finite complement topology \Tau_{f} ( a set X) : Collection of all subsets U of X s.t. X-U either is finite or is all of X.
2.1.5 Suppose \Tau and \Tau’ are two topologies on a given set X.
2.1.5.1 \Tau’ is finer than \Tau : \Tau \subset \Tau’
2.1.5.2 \Tau’ is strictly finer than \Tau : \Tau \subsetneq \Tau’
2.1.5.3 \Tau’ is coarser than \Tau : \Tau’ \subset \Tau
2.1.5.4 \Tau’ is strictly coarser than \Tau : \Tau’ \subsetneq \Tau
2.1.5.5 \Tau is comparable with \Tau’ : \Tau’ \subset \Tau or \Tau \subset \Tau’
2.2 Basis for a Topology
2.2.1 Basis for a topology on X (a set X) : a collection \mathcal{B} of subsets of X (called basis elements) s.t.
2.2.1.1 For each x \in X, there is at least one basis element B containing x.
2.2.1.2 If x belongs to the intersection of two basis elements B_{1} and B_{2}, then there is a basis element B_{3} containing x s.t. B_{3} \subset B_{1} \cap B_{2}.
2.2.2 Topology \Tau generated by \mathcal{B}
2.2.2.1 Open in X ( a subset U of X) : for each x \in U, there is a basis element B \in \mathcal{B} s.t. x \in B and B \subset U.
2.2.3 (Lem 13.1) : Let X be a set; let \mathcal{B} be a basis for a topology \Tau on X. then \Tau equals the collection of all unions of elements of \mathcal{B}.
2.2.4 (Lem 13.2) : Let X be a topological space. Suppose that \mathcal{C} is a collection of open sets of X s.t. for each open set U of X and each x in U , there is an element C of \mathcal{C} s.t. x \in C \subset U. Then \mathcal{C} is a basis for the topology of X.
2.2.5 (Lem 13.3) Let \mathcal{B} and \mathcal{B} be bases for the topologies \Tau and \Tau’ respectively on X. Then the following are equivalent.
2.2.5.1 \Tau’ is finer than \Tau.
2.2.5.2 For each x \in X and each basis element B \in \mathcal{B} containing x, there is a basis element B’ \in \mathcal{B}’ s.t. x \in B’ \subset B.
2.2.6 Standard topology on the real line : Topology generated by \mathcal{B}
2.2.6.1 \mathcal{B} is the collection of all open intervals in the real line, (a,b) = {x | a < x < b}.
2.2.7 Lower limit topology \mathbb{R}_{l} on \mathbb{R} : Topology generated by \mathcal{B}’.
2.2.7.1 \mathcal{B}’ is the collection of all half-open intervals
2.2.8 K-topology \mathbb{R}_{K} on \mathbb{R} : Topology generated by \mathcal{B}’’
2.2.8.1 \mathcal{B}’’ is the collection of all open intervals (a,b) , along with all sets of the form (a,b) – K
2.2.8.1.1 K = {x | x = 1/n, for n \in \mathbb{Z}_{+}}
2.2.9 (Lem 13.4) Topologies of \mathbb{R}{l} and \mathbb{R}{K} are strictly finer than the standard topology on \mathbb{R}, but not comparable with one another.
2.2.10 Subbasis \mathcal{S} for a topology on X : a collection of subsets of X whose union equals X
2.2.10.1 Topology generated by the subbasis \mathcal{S} : the collection \Tau of all unions of finite intersections of elements of \mathcal{S}
2.3 Order topology
2.3.1 Order topology : If X is a simply ordered set, there is a standard topology for X, defined using order relation.
2.3.2 Suppose X is a set having a simple order relation <. Given elements a and b of X s.t. a <b,
2.3.2.1 Open interval (a,b) = {x| a<x<b}
2.3.2.2 Closed interval [a,b] = {x | a \le x \le b}
2.3.2.3 Half-open intervals]
2.3.2.3.1 (a,b]= {x | a < x \le b }
2.3.2.3.2 [a,b) = {x | a \le x < b}
2.3.3 Order topology : Let X be a set with simple order relation, assume X has more than one element. Let \mathcal{B} be the collection of all sets of the following types. The coillection \mathcal{B} is a basis for a topology on X, which is order topology.
2.3.3.1 All open intervals in X
2.3.3.2 All intervals of the form [a_{0},b) , where a_{0} is the smallest element (if any) of X.
2.3.3.3 All intervals of the form (a, b_{0}], where b_{0} is the largest element (if any) of X.
2.3.4 If X is an ordered set and a is an element of X, there are four subsets of X that are called rays determined by a.
2.3.4.1 Open rays
2.3.4.1.1 (a, +\infty) = {x | x > a}
2.3.4.1.2 (-\infty, a) = {x | x < a}
2.3.4.2 Closed rays
2.3.4.2.1 [a, + \infty) = {x | x \ge a}
2.3.4.2.2 (-\infty, a] = {x | x \le a}
2.3.4.3 A topology generated using open rays as a subbasis contains the order topology
2.4 Product Topology on X \times Y
2.4.1 Product topology on X \times Y ( topological spaces X and Y) : topology having as basis the collection \mathcal{B} of all sets of the form U \times V, where U is an open subset of X and V is an open subset of Y.
2.4.2 (Thm 15.1) If \mathcal{B} is a basis for the topology of X and \mathcal{C} is a basis for the topology of Y, then the collection \mathcal{D} = {B \times C | B \in \mathcal{B} and C \in \mathcal{C}} is a basis for the topology of X \times Y.
2.4.2.1 Pf) (Lem 13.2)
2.4.3 Projections of X \times Y onto its first and second factors
2.4.3.1 \pi_{1} : X \times Y -> X , \pi_{1} (x,y) = x
2.4.3.2 \pi_{2} : X \times Y -> Y, \pi_{2} (x,y) = y
2.4.4 (Thm 15.2) The collection \mathcal{S} = {\pi^{-1}{1}(U) | U open in X} \cup {\pi^{-1}{2}(V) | V open in Y} is a subbasis for the product topology on X \times Y.
2.5 Subspace topology
2.5.1 Subspace topology : Let X be a topological space with topology \Tau. If Y is a subset of X, the collection \Tau_{Y} = {Y \cap U | U \in \Tau} is a topology on Y.
2.5.1.1 Subspace of X : Y with this topology
2.5.2 (Lem 16.1) If \mathcal{B} is a basis for the topology of X then the collection \mathcal{B}_{Y} = {B \cap Y | B \in \mathcal{B}} is a basis for the subspace topology on Y.
2.5.2.1 Pf) (Lem 13.2)
2.5.3 Open in Y ( a set U ) : U belongs to the topology of Y.
2.5.4 (Lem 16.2) Let Y be a subspace of X. If U is open in Y and Y is open in X, then U is open in X.
2.5.5 (Thm 16.3) If A is a subspace of X and B is a subspace of Y, then the product topology on A \times B is the same as the topology A \times B inherits as a subspace of X \times Y.
2.5.6 Ordered square I^{2}_{0} : set I \times I in the dictionary order topology (I = [0,1])
2.5.6.1 Dictionary order topology of it \neq subset topology inherited from \mathbb{R}^{2}
2.5.7 Convex in X (a subset Y of an ordered set X) : for each pair of points a<b of Y, the entire interval (a,b) of points of X lies in Y.
2.5.8 (Thm 16.4) : Let X be an ordered set in the order topology; Let Y be a subset of X that is convex in X, Then the order topology on Y is the same as the topology Y inherits as a subspace of X.]
2.6 Closed sets and limit points
2.6.1 Closed
2.6.1.1 Closed ( a subset A of a topological space X) : if the set X-A is open
2.6.1.2 (Thm 17.1) Let X be a topological space, then the following conditions hold
2.6.1.2.1 \empty and X are closed.
2.6.1.2.2 Arbitrary intersections of closed sets are closed
2.6.1.2.3 Finite unions of closed sets are closed.
2.6.1.3 Closed in Y ( a set A) : If Y is a subspace of X, if A is a subset of Y and if A is closed in the subspace topology of Y.
2.6.1.4 (Thm 17.2) Let Y be a subspace of X. Then a set A is closed in Y iff it equals the intersection of a closed set of X with Y.
2.6.1.5 (Thm 17.3) Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.
2.6.2 Closure and Interior of a set
2.6.2.1 Interior ( a subset A of a topological space X) : union of all open sets contained in A
2.6.2.2 Closure ( a subset A of a topological space) : intersection of all closed sets containing A.
2.6.2.3 Int A \subset A \subset \bar{A}
2.6.2.4 (Thm 17.4) Let Y be a subspace of X. Let A be a subset of Y. Let \bar{A} denote the closure of A in X. Then the closure of A in Y equals \bar{A} \cap Y.
2.6.2.4.1 Pf) Thm 17.2
2.6.2.5 Intersects (sets A, B) : the intersection A \cap B is not empty.
2.6.2.6 (Thm 17.5) Let A be a subset of the topological space X.
2.6.2.6.1 x \in \bar{A} iff every open set U containing x intersects A.
2.6.2.6.2 Suppose the topology of X is given by a basis, then x \in \bar{A} iff every basis element B containing x intersects A.
2.6.2.7 U is a neighborhood of x (a set U, an element x) : U is an open set containing x
2.6.3 Limit points
2.6.3.1 Limit point x of A ( a point x in X , a subset A of the topological space X) : every neighborhood of x intersects A in some point other than x itself.
2.6.3.1.1 x is a limit point of A if it belongs to the closure A – {x}
2.6.3.2 (Thm 17.6) Let A be a subset of the topological space X. Let A’ be the set of all limit points of A. Then \bar{A} = A \cup A’.
2.6.3.2.1 Pf) Thm 17.5.
2.6.3.2.2 (Cor) A subset of a topological space is closed iff it contains all its limit points.
2.6.4 Hausdorff spaces
2.6.4.1 Converges to the point x of X ( a sequence x_{1}, x_{2}, … of the points of the space X) : corresponding to each neighborhood U of x, there is a positive integer N s.t. x_{n} \in U for all n \ge N.
2.6.4.2 Hausdorff space ( a topological space X) : for each pair x_{1}, x_{2} of distinct points of X, there exists neighborhoods U_{1}, U_{2} of x_{1} and x_{2}, respectively, that are disjoint.
2.6.4.3 (Thm 17.8) Every finite point set in a Hausdorff space X is closed.
2.6.4.4 T_{1} axiom : Every finite point sets in a space are closed
2.6.4.5 (Thm 17.9) Let X be a space satisfying the T1 axiom. Let A be a subset of X. Then the point x is a limit point of A iff every neighborhood of x contains infinitely many points of A.
2.6.4.6 (Thm 17.10) If X is a Hausdorff space, then a sequence of points of X converges to at most one point of X.
2.6.4.6.1 x_{n} -> x <= Limit x of the sequence x_{n}
2.6.4.7 (Thm 17.11) Every simply ordered set is a Hausdorff space in the order topology. The product of two Hausdorff spaces is a Hausdorff space. A subspace of a Hausdorff space is a Hausdorff space.
2.7 Continuous functions
2.7.1 Continuity of a Function
2.7.1.1 Continuous ( A function f: X ->Y, Topological spaces X , Y) : For each open subset V of Y, the set f^{-1}(V) is an open subset of X.
2.7.1.1.1 F is continuous relative to specific topologies on X and Y.
2.7.1.2 (Thm 18.1) Let X and Y be topological spaces. Let f : X ->Y. then the following are equivalent.
2.7.1.2.1 f is continuous.
2.7.1.2.2 For every subset A of X, one has f(\bar{A}) \subset \bar{f(A)}.
2.7.1.2.3 For every closed set B of Y, the set f^{-1}(B) is closed in X.
2.7.1.2.4 For each x \in X and each neighborhood V of f(x), there is a neighborhood U of x s.t. f(U) \subset V.
2.7.1.2.4.1 f is continuous at the point x.
2.7.2 Homeomorphism
2.7.2.1 Homeomorphism ( f : X -> Y, topological spaces X, Y) : Let f be bijection. Both the function and the inverse function f^{-1} : Y ->X are continuous,
2.7.2.2 Topological property of X : any property of X expressed in terms of the topology of X yields, via the correspondence f, the corresponding property for the space Y.
2.7.2.3 Topological imbedding f of X in Y ( topological spaces X and Y, f : X->Y injective) : the function f’ : X -> f(X) , which is bijective, happens to be homeomorphism
2.7.2.4 Unit circle s^{1} = { x \times y | x ^{2} + y^{2} = 1} , [0,1)
2.7.2.4.1 F : [0,1) -> s^{1} , (cos 2\pi t, sin 2\pi t)
2.7.3 Constructing Continuous functions
2.7.3.1 (Rules for constructing continuous functions)(Thm 18.2) : Let X, Y, Z be topological spaces.
2.7.3.1.1 (Constant function) If f : X->Y maps all of X into the single point y_{0} of Y, then f is continuous.
2.7.3.1.2 (Inclusion) If A is a subspace of X, the inclusion function j : A ->X is continuous.
2.7.3.1.3 (Composites) If f : X->Y and g : Y ->Z are continuous, then the map g \bullet f : X -> Z is continuous.
2.7.3.1.4 (Restricting the domain) If f : X->Y is continuous, and if A is a subspace of X, then the restricted function f | A : A->Y is continuous.
2.7.3.1.5 (Restricting or expanding the range) Let f : X -> Y be continuous. If Z is a subspace of Y containing the image set f(X), then the function g : X ->Z obtained by restricting the range of f is continuous. If Z is a space having Y as a subspace, then the function h: X ->Z obtained by expanding the range of f is continuous.
2.7.3.1.6 (Local formulation of continuity) The map f: X->Y is continuous if X can be written as the union of open sets U_{\alpha} s.t. f | U_{\alpha} is continuous for each \alpha.
2.7.3.2 (The pasting lemma)(Thm 18.3) : Let X = A \cup B, where A and B are closed in X. Let f : A -> Y and g : B -> Y be continuous. If f (x) = g (x) for every x \in A \cap B, then f and g combine to give a continuous function h : X -.Y, defined by setting h(x) = f(x) if x \in A, and h(x) = g(x) if x \in B.
2.7.3.3 (Map into products) (Thm 18.4) Let f : A -> X \times Y be given by the equation f(a) = (f_{1}(a), f_{2}(a)). Then f is continuous iff the function f_{1} : A -> X and f_{2} : A -> Y are continuous.
2.7.3.3.1 Maps f_{1} and f_{2} are called the coordinate functions of f.
2.7.3.4 (Uniform Limit Theorem on a Real space) : If a sequence of continuous real-valued functions of a real variable converges uniformly to a limit function, then the limit function is necessarily continuous.
2.8 Product Topology
2.8.1 J-tuple of elements of X ( an Index set J, a set X) : a function \mathbf{x} : J -> X
2.8.1.1 \alpha th coordinate x_{\alpha} of \mathbf{x} : the value of \mathbf{x} at \alpha
2.8.1.2 (x_{\alpha})_{\alpha \in J} \mathbf{x}
2.8.2 Cartesian product \prod_{\alpha \in J} A_{\alpha} ( an indexed family {A_{\alpha}}{\alpha \in J} ) : a set of all J-tuples (x{\alpha}){\alpha \in J} of elements of X s.t. x{\alpha} \in A_{\alpha} for each \alpha \in J.
2.8.2.1 Set of all functions \mathbf{x} : J -> \bigcup_{\alpha \in J} A_{\alpha} s.t \mathbf{x} (\alpha) \in A_{\alpha} for each \alpha \in J.
2.8.3 Box topology( {X_{\alpha}}_{\alpha \in J}) : the topology generated by this basis
2.8.3.1 Basis for a topology on the product space \prod_{\alpha \in J} X_{\alpha} the collection of all sets of the form \prod_{\alpha \in J} U_{\alpha} where U_{\alpha} is open in X_{\alpha}.
2.8.3.2 Projection mapping associated with the index \beta : \pi_{\beta}((x_{\alpha}){\alpha \in J}) = x{\beta}
2.8.4 Product topology : The topology generated by the subbasis \mathcal{S}
2.8.4.1 \mathcal{S} = \bigcup_{\beta \in J} \mathcal{S}_{\beta}
2.8.4.1.1 \mathcal{S}{\beta} = {\pi^{-1}{\beta}(U_{\beta}) | U_{\beta} open in X_{beta}}
2.8.4.2 Product space : \prod_{\alpha \in J} X_{\alpha}
2.8.5 (Comparison of the box and product topologies) (Thm 19.1) : The box topology on \prod X_{\alpha} has as basis all sets of the form \prod U_{\alpha}, where U_{\alpha} is open in X_{\alpha} for each \alpha . The product topology on \prod X_{\alpha} has as basis all sets of the form \prod U_{\alpha} , where U_{\alpha} is open in X_{\alpha} for each \alpha and U_{\alpha} equals X_{\alpha} except for finitely many values of \alpha.
2.8.5.1 Box topology is finer than the product topology, but pretty much the same.
2.8.5.2 When considering the product \prod X_{\alpha}, the product topology is usually assumed.
2.8.6 (Thm 19.2) Suppose the topology on each space X_{\alpha} is given by a basis \mathcal{B}{\alpha}. The collection of all sets of the form \prod{\alpha \in J} B_{\alpha} where B_{\alpha} \in \mathcal{B}{\alpha} for each \alpha, will serve as a basis for the box topology on \prod{\alpha \in J} X_{\alpha}.
2.8.6.1 The collection of all sets of the same form, where B_{\alpha} \in \mathcal{B}{\alpha} for finitely many indices \alpha and B{\alpha} = X_{\alpha} for all the remaining indices, will serve as a basis for the product topology \prod_{\alpha \in J} X_{\alpha}.
2.8.7 (Thm 19.3) Let A_{\alpha} be a subspace of X_{\alpha}. For each \alpha \in J. Then \prod A_{\alpha} is a subspace of \prod X_{\alpha} if both products are given the box topology, or if both products are given the product topology.
2.8.8 (Thm 19.4) If each space X_{\alpha}is a Hausdorff space, then \prod X_{\alpha} is a Hausdorff space in both the box and product topologies.
2.8.9 (Thm 19.5) Let {X_{\alpha}} be an indexed family of spaces ; let A_{\alpha} \subset X_{\alpha} for each \alpha. If \prod X_{\alpha} is given either the product or the box topology, then \prod \bar{A_{\alpha}} = \bar{\prod A_{\alpha}}.
2.8.10 (Thm 19.6) Let f: A => \prod_{\alpha \in J} X_{\alpha} be given by e equation f(a) = (f_{\alpha} (a) ) {\alpha \in J} where f{\alpha} : A -> X_{\alpha} for each \alpha . Let \prod X_{\alpha} have the product topology, then the function f is continuous iff each function f_{\alpha} is continuous.
2.9 Metric topology
2.9.1 Metiric on a set X : a function d : X \times X -> R having the following properties.
2.9.1.1 d(x,y) \ge 0 for all x, y \in X; equality iff x = y.
2.9.1.2 d(x,y) = d(y,x) for all x,y \in X.
2.9.1.3 (Triangle inequality) d(x,y) + d(y,z) \ge d(x,z) for all x,y,z \in X.
2.9.2 Distance between x and y : d(x,y)
2.9.2.1 B_{d} (x,\epsilon) : \epsilon -ball centered at x
2.9.3 Metric topology induced by d ( a metric d on the set X) : Collection of all \epsilon -balls B_{d} (x,\epsilon) for x \in X and \epsilon > 0 being a basis for a topology on X.
2.9.4 Metrizable (a topological space X) : there exists a metric d on the set X that induces the topology of X.
2.9.5 Metric space : a metrizable space X together with a specific metric d that gives the topology of X.
2.9.6 Bounded ( a subset A of a metric space X with metric d) : there is some number M s.t. d(a_{1}, a_{2}) \le M for every pair a_{1}, a_{2} of points of A.
2.9.6.1 Diameter of A : if A is bounded and nonempty, diam A = sup{d(a_{1}, a_{2}) | a_{1}, a_{2} \subset A}
2.9.7 Standard bounded metric corresponding to d ( a metric space X with metric d) : \bar{d} : X \times X -> \mathbb{R} by the equation \bar{d} (x,y) = min{d(x,y) , 1}. Then \bar{d} is a metric that induces the same topology as d.
2.9.8 Norm of \mathbf{x} (\mathbf{x} = (x_{1}, …, x_{n} ) in \mathbb{R}^{n} ) : \Vert x \Vert = ( x^{2}{1} + … + x^{2}{n} )^{frac{1}{2}}
2.9.8.1 Euclidean metric d on \mathbb{R}^{n} : d(x,y) = \Vert \mathbf{x} - \mathbf{y} \Vert = [(x_{1} – y_{1})^{2} + … + (x_{n} – y_{n})^{2}]^{frac{1}/{2}}
2.9.8.2 Squate metric \rho : \rho (\mathbf{x}, \mathbf{y}) = max{ \vert x_{1} – y_{1} \vert , …, \vert x_{n} – y_{n} \vert }
2.9.9 (Lem 20.2) Let d and d’ be two metrics on the set X; Let \Tau and \Tau’ be the topologies they induce, respectively. Then \Tau’ is finer than \Tau iff for each x in X and each \epsilon >0, there exists a \delta >0 s.t. B_{d’} (x,\delta) \subset B_{d} (x,\epsilon).
2.9.9.1 Pf) (Lem 13.3)
2.9.10 (Thm 20.3) The topologies on \mathbb{R}^{n} induced by the Euclidean metric d and the squate metric \rho are the same as the product topology on \mathbb{R}^{n}.
2.9.11 Uniform metric on \mathbb{R}^{J} ( an index set J) : Given points \mathbf{x} = (x_{\alpha}){\alpha \in J} and \mathbf{y} = (y{\alpha}){\alpha \in J} of \mathbb{R}^{J}. Define a metric \bar{\rho} on \mathbb{R}^{J} by the equation \bar{\rho} (\mathbf{x} , \mathbf{y}) = sup{\bar{d} (x{\alpha} , y_{\alpha}) | \alpha \in J}, where \bar{d} is the standard bounded metric on \mathbb{R}.
2.9.11.1 Uniform topology : Topology \bar{\rho} induces
2.9.12 (Thm 20.4) Uniforn topology on \mathbb{R}^{J} is finer than the product topology and coarser than the box topology; these three topologies are all different if J is infinite.
2.9.13 (Thm 20.5) Let \bar{d} (a,b) = min{\vert a-b \vert , 1} be the standard bounded metric on \mathbb{R}. if \mathbf{x} and \mathbf{y} are two points of \mathbb{R}^{\omega}, define D(x,y) = sup{frac{\bar{d} (x_{i}, y_{i})} {i}}. Then D is a metric that induces the product topology on \mathbb{R}^{\omega}.
2.10 Metric topology continued
2.10.1 (Thm 21.1) Let f : X->Y; let X and Y be metrizable with metrics d_{X} and d_{Y}, respectively. Then continuity of f is equivalent to the requirement that given x \in X and given \epsilon > 0, there exists \delta > 0 s.t. d_{X} (x,y) < \delta -> d_{Y} (f_{x}, f_{y}) < \epsilon.
2.10.2 (Sequenece Lemma) (Lem 21.2) : Let X be a topological space; let A \subset X. if there is a sequence of points of A converging to x, then x \in \bar{A}; the converse holds if X is metrizable.
2.10.2.1 Pf) Thm 17.5
2.10.3 (Thm 21.3) Let f : X -> Y. If the function f is continuous, then for every convergent sequence x_{n} -> x in X, the sequence f(x_{n}) converges to f(x). the converse holds if X is metrizable.
2.10.3.1 Pf) Sequence lemma
2.10.4 a countable basis at the point x ( a space X) : there is a countable collection {U_{n}}{n \in \mathbb{Z}{+}} of neighborhoods of x s.t. any neighborhood U of x contains at least one of the sets U_{n}.
2.10.5 First countability axiom : A space X that has a countable basis at each of its points
2.10.6 (Lem 21.4) Addition, subtraction, and multiplication operations are continuous functions from \mathbb{R} \times \mathbb{R} into \mathbb{R}, and the quotient operation is a continuous function from \mathbb{R} \times (\mathbb{R} – {0}) into \mathbb{R}.
2.10.7 (Thm 21.5) If X is a topological space, and if f,g : X -> \mathbb{R} are continuous functions, then f + g, f – g, and f \cdot g are continuous. If g(x) \neq 0 for all x, then f/g is continuous.
2.10.7.1 Pf) (Thm 18.4)
2.10.8 Converges uniformly to the function f: X -> Y ( f_{n} : X -> Y a sequence of functions from the set X to the metric space Y) : Let d be the metric for Y. Given \epsilon >0, there exists an integer N s.t. d(f_{n} (x) , f (x)) < \epsilon for all n > N and all x in X;
2.10.9 (Uniform limit theorem) (Thm 21.6) : Let f_{n} : X -> Y be a sequence of continuous functions from the topological space X to the metric space Y. if (f_{n}) converges uniformly to f, then f is continuous.
2.10.10 Examples of spaces not metrizable
2.10.10.1 \mathbb{R}^{\omega} in the box topology is not metrizable.
2.10.10.2 \An uncountable product of \mathbb{R} with itself is not metrizable.
2.11 Quotient topology
2.11.1 Quotient map ( The map p) : Let X and Y be topological spaces; let p : X -> Y be a surjective map. If, A subset U of Y is open in Y iff p^{-1}(U) is open in Y, p is a quotient map.
2.11.2 Saturated with respect to the surjective map p : X -> Y ( a subset C of X) : C contains every set p^{-1}({y}) that it intersects.
2.11.3 If p : X->Y is a surjective continuous map that is either open or closed, then p is a quotient map.
2.11.3.1 Open map (a map f : X -> Y) : for each open set U of X the set f(U) is open in Y
2.11.3.2 Closed map (a map f : X -> Y) : for each closed set A of X the set f(A) is closed in Y
2.11.4 Quotient topology induced by p : If X is a space and A is a set and if p : X -> A is a surjective map, then there exists exactly one topology \Tau on A relative to which p is a quotient map.
2.11.5 Quotient space of X (a topological space X) : Let X* be a partition of X into disjoint subsets whose union is X. Let p : X -> X* be the surjective map that carries each point of X to the element of X* containing it. In the quotient topology induced by p, the space X* is called a quotient space of X.
2.11.6 (Thm 22.1) Let p : X->Y be a quotient map; Let A be a subspace of X that is saturated with respect to p; let q : A -> p(A) be the map obtained by restricting p.
2.11.6.1 If A is either open or closed in X, then q is a quotient map.
2.11.6.2 If p is either an open map or a closed map, then q is a quotient map.
2.11.7 (Thm 22.2) Let p : X -> Y be a quotient map. Let Z be a space and let g : X -> Z be a map that is constant on each set \rho^{-1}({y}), for y \in Y. Then g induces a map f : Y -> Z s.t. f \bullet p = g. The induced map f is continuous iff g is continuous; f is a quotient map iff g is a quotient map.
2.11.7.1 (Cor) Let g : X ->Z be a surjective continuous map. Let X* be the following collection of subsets of X : X* = {g^{-1}({z}) | z \in Z}. Give X* the quotient topology.
2.11.7.1.1 The map g induces a bijective continuous map f: X* -> Z, which is a homeomorphism iff g is a quotient map.
2.11.7.1.2 If Z is Hausdorff, so is X*.