Topology Test 1

Homeomorphisms

- Let X and Y be topological spaces. Let f:X→Y be a bijection. If both the function f and the inverse function f^-1:Y→X are continuous, then f is called a homeomorphism.

Finite complement topology

- Let X be a set, T the collection of all subsets U of X such that X-U is either finite or all of X. Then T is the finite complement topology.

Subspace topology

- Let X be a topological space with topology T. If Y is a subset of X, the collection Ty = {Y∩U|U∈T} is the subspace topology. - Y is called a subspace of X.

Product topology

- Let X,Y be topological spaces. The product topology on XxY is the topology having as a basis the collection ß of all sets of the form UxV, where U is open in X and V is open in Y.

Arbitrary unions and intersections

- Unions: ∪A = {x|x∈A for at least 1 A}. - Intersections: ∩A = {x|x∈A for every A}.

w-tuple

- Given a set X, we define a w-tuple of elements of X to be a function x:Z+→X. - We also call such a function a sequence, or an infinite sequence, of elements of X.

Interior

- Given a subset A of a topological space X, the interior of A is defined as the union of all open sets contained in A. - Denoted intA, and it is an open set. - IntA⊂A⊂Abar. - A is open, A=intA.

Equivalence class

- Given an equivalence relation ~ on a set A, and an element x of A, we define the equivalence class determined by x, a subset E of A, to be E={y|y~x}.

Composite

- Given functions f:A→B and g:B→C, we define the composite g⁰f of f and g as the function g⁰f:A→C defined by the equation (g⁰f)(a) = g(f(a)).

ε-ball

- Given ε>0, the set Bd(X,ε) = {y|d(x,y)<ε} of all points y whose distance from x is less than ε is called the ε-ball (centered at x).

Limit points (cluster point or point of accumulation)

- If A is a subset of the topological space X and if x is a point of X, we say that x is a limit point of A if every neighborhood of x intersects A in some point other than x itself. - x is a limit point of A if it belongs to the closure of A-{x}.

Every finite point in a Hausdorff space X

- Is closed.

If A is a subspace of X and B a subspace of Y, then the product topology on AxB is

- The same as the topology AxB inherits as a subspace of XxY.

If X is an ordered set, and a is an element of X, there are 4 subsets of X that are called rays determined by a. They are:

(1) (a,+∞) = {x|x>a}. (2) (-∞,a) = {x|x<a}. (3) [a,+∞) = {x|x≥a}. (4) (-∞,a] = {x|x≤a}. Where 1,2 are open rays and 3,4 are closed rays.

Let X be a set with a simple order relation; assume X has more than one element. Let ß be the collection of all sets of the following types:

(1) All open intervals (a,b) in X. (2) All intervals of the form [a₀,b) where a₀ is the smallest element (if any) of X. (3) All intervals of the form (a,b₀] where b₀ is the largest element (if any) of X. - The collection ß is a basis for a topology on X, which is called the order topology.

A set is countable if B is a nonempty set and TFAE:

(1) B is countable. (2) There is a surjective function f:Z+→B. (3) There is an injective function g:B→Z+.

A relation C on a set A is called an order relation if it has the following properties:

(1) Comparability: For every x,y in A for which x≠y, either xCy or yCx. (x<y or y<x). (2) Non-reflectivity: For no x in A does the relation xCx hold. (3) Transitivity: If xCy and yCz, then xCz. (x<y and y<z then x<z). - < is the symbol for order relations.

If X is a set, a basis for a topology on X is a ß of subsets of X (called basis elements) such that

(1) For each x∈X, there is at least one basis element B such that x∈B. (2) If x∈B₁∩B₂, then there is a B₃ such that x∈B₃ and B₃⊂B₁∩B₂.

Let ß and ß' be bases for the topologies T and T' on X. Then TFAE

(1) T' is finer than T. (2) For each x∈X and each basis element B∈ß containing x, there is a basis element B'∈ß such that x∈B'⊂B.

Let A be a subset of the topological space X. Then

(1) Then x∈A° iff every open set U containing x intersects A. (2) Supposing the topology of X is given by a basis, then x∈A° iff every basis element b containing x intersects A.

A metric on a set X is a function d:XxX→R with the following properties

(1) d(x,y)≥0 for all x,y∈X. Equality holds iff x=y. (2) d(x,y) = d(y,x) for all x,y∈X. (3) (Triangle inequality) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z∈X.

Let X and Y be topological spaces. Let f:X→Y. Then TFAE

(1) f is continuous. (2) For every subset A of X, one has f(A°)⊂(f(A))°. (3) For every closed set B of Y, the set f^-1(B) is closed in X. (4) For each x∈X and each neighborhood V of f(x), there is a neighborhood U of x such that f(U)⊂V. - If 4 holds for every x in X, then f is continuous at the point x.

Let X be a topological space. Then the following hold:

(1) ∅ and X are closed. (2) Arbitrary intersections of closed sets are closed. (3) Finite unions of closed sets are closed.

Binary Operation

- A binary operation on a set A is a function f mapping AxA into A.

Injective, surjective, bijective

- A function f:A→B is said to be injective (1-1) if for each pair of distinct points of A, their images under f are distinct. It is said to be surjective (onto) if every element of B is the image of some element of A under the function f. If f is both 1-1 and onto, then it is said to be bijective (1-1 correspondence). - 1-1: f(a)=f(a') → a=a' (depends only on the rule of f). - Onto: b∈B → b=f(a) for a least one a∈A (depends on the range of f as well).

Function

- A function is a rule of assignment r, together with a set B that contains the image of r. The domain A of the rule r is also called the domain of the function f, the image set of r is also called the image set of f, and the set B is called the range of f.

Partition

- A partition of a set A is a collection of disjoint nonempty subsets of A whose union is all of A.

Relation

- A relation on a set A is a subset C of the cartesian product AxA. - Use the notation xCy to mean the same thing as (x,y)∈C, read as x is in the relation C to y. - Can also be denoted by ~, so xCx = x~x.

Infinite, countably infinite

- A set A is said to be infinite if it is not finite. - It is said to be countably infinite if there is a bijective correspondence f:A→Z+.

Countable, uncountable

- A set is said to be countable if it is either finite or countably infinite. - A set that is not countable is said to be uncountable.

Finite sets

- A set is said to be finite if there is a bijective correspondence of A with some section of the positive integers. That is, A is finite if it is empty, or if there is a bijection f:A→{1,...,n} for some positive integer n. - In the former case, we say that A has cardinality 0, in the latter case, we say that A has cardinality n.

Subbasis

- A subbasis S for a topology on X is a collection of subsets of X whose union equals X. The topology generated by the subbasis S is defined the be the collection T of all unions of finite intersections of elements of S.

Closed sets

- A subset A of a topological space X is said to be closed if the set X-A (the complement) is open.

The topology T generated by ß is

- A subset U of X said to be open in X if for each x∈U, there is a basis element B∈ß such that x∈B and B⊂U.

Hausdorff Space

- A topological space X is called a Hausdorff space if for each pair x₁,x₂ of distinct points of X, there exist neighborhoods U₁ and U₂ of x₁ and x₂ that are disjoint.

Topology

- A topology on a set X is a collection T of subsets of X having the following properties: (1) ∅ and X are in T. (2) The union of the elements of any subcollection of T is in T. (3) The intersection of the elements of any finite subcollection of T is in T. - A set X for which a topology T has been specified is called a topological space.

Equivalence relation

- An equivalence relation on a set A is a relation C on A having the following properties: (1) Reflexivity: xCx ∀x∈A. (2) Symmetry: If xCy then yCx. (3) Transitivity: If xCy and yCz, then xCz.

Least upper bound property, greatest lower bound property

- An ordered set A is said to have the least upper bound property if every nonempty subset A₀ of A that is bounded above has a least upper bound. - The set A is said to have the greatest lower bound property if every nonempty subset A₀ of A that is bounded below has a greatest lower bound.

Empty set

- A∪∅=A and A∩∅=∅.

Logic: If P then Q

- Contrapositive: If Q is not true, then P is not true. - Converse: If Q then P. - Negate: for every x∈A, P holds into for at least one x∈A, P does not hold.

If X is a Hausdorff space, then a sequence of points of X

- Converges to at most one point of X.

A countable union of countable sets is

- Countable.

A finite product of countable sets is

- Countable.

A subset of countable sets is

- Countable.

Distributive and DeMorgan's:

- Distributive: A∪(B∩C) = (A∪B)∩(A∪C). - DeMorgan's: (i) A-(B∪C) = (A-B)∩(A-C). (ii) A-(B∩C) = (A-B)∪(A-C).

Domain and range

- Domain r = {c|∃ d∈D s.t. (c,d)∈r}. - Image r = {d|∃ c∈C s.t. (c,d)∈r}.

Two equivalence classes E and E' are

- Either disjoint or equal.

Let X be a set, and ß a basis for the topology T on X. Then T

- Equals the collection of all unions of elements of ß.

Well ordering property

- Every nonempty subset of Z+ has a smallest element.

Finite unions and finite cartesian products of finite sets are

- Finite.

Open interval, immediate predecessor, immediate successor

- If X is a set and < is an order relation on X, and if a<b, we use the notation (a,b) to denote the set {x|a<x<b} is called an open interval in X. - If this set is empty, we call a the immediate predecessor of b, and we call b the immediate successor of a.

Open set

- If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the subcollection T.

Metrizable

- If X is a topological space, X is said to be metrizable if there exists a metric d on the set X that induces the topology of X. - A metric space is a metrizable space X together with a specific metric d that gives the topology of X.

Discrete and trivial topology

- If X is any set, the collection of all subsets of X is a topology on X called the discrete topology. - The collection consisting of X and ∅ only is a topology on X called the indiscrete topology, or the trivial topology.

Metric topology

- If d is a metric on the set X, then the collection of all ε-balls Bd(X,ε) for x∈X and ε>0, is a basis for a topology on X, called the metric topology induced by d.

Inverse

- If f is bijective, then there exists a function from B to A called the inverse of f. - Denoted f^-1, defined by letting f^-1(b) be that unique element a of A for which f(a)=b.

Restriction

- If f:A→B and if A₀ is a subset of A, we define the restriction of f to A₀ to be the function mapping A₀ into B whose rule is {(a, f(a))|a∈A₀}. - It is denoted by f|A₀, which is read "f restricted to A₀."

Standard topology

- If ß is the collection of all open intervals in the real line, (a,b) = {x|a<x<b}, the topology generated by ß is called the standard topology on the real line.

Lower limit topology

- If ß' is the collection of all half-open intervals of the form [a,b) = {x|a≤x<b}, where a<b, the topology generated by ß' is called the lower limit topology on R.

A rule of assignment

- Is a subset r of the cartesian product C×D (C×D = {(a,b)|a∈A and b∈B}) of two sets, having the property that each element of C appears as the first coordinate of at most one ordered pair belonging to r. - A subset r of C×D is a rule of assignment if [(c,d)∈r and (c,d')∈r] → [d=d'].

If C is an infinite subset of Z+, then C

- Is countably infinite.

The set Z+xZ+

- Is countably infinite.

A subset of a topological space is closed iff

- It contains all of its limit points.

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.

J-tuple, α-th coordinate.

- Let J be an index set. Given a set X, we define a J-tuple of elements of X to be a function x:J→X. If α is an element of J, we call it the α-th coordinate of x.

K-topology

- Let K denote the set of all numbers of the form 1/n, and ß'' the collection of all open intervals (a,b) along with all sets of the form (a,b)-K. Then the topology generated by ß'' is the K-topology on R.

Product topology, product space

- Let S_ß denote the collection S_ß = {π_ß inverse (U_ß)|U_ß open in X_ß}. - Let S denote the union of these collections, S = ∪S_ß. - The topology generated by the subbasis S is called the product topology, and ΠXα is called a product space.

Continuous

- Let X and Y be topological spaces. A function f:X→Y is said to be continuous if for each open subset V of Y, the set f^-1(V) is an open subset of X.

Pasting lemma

- Let X=A∪B where A and B are cloesd in X. Let f:A→Y and g:B→Y be continuous. If f(x)=g(x) for every x∈A∩B, then f and g combine to give a continuous function h:X→Y, defined by setting h(x)=f(x) if x∈A, and h(x)=g(x) if x∈B.

Image, pre-image

- Let f:A→B. If A₀ is a subset of A, we denote by f(A₀) the set of all images of points of A₀ under the function f; this set is called the image of A₀ under f. - Formally: image f(A₀) = {b|b=f(a) for at least 1 a∈A₀}. - If B₀ is a subset of B, we denote f^-1(B₀) the set of all elements of A whose images under f lie in B₀; it is called the pre-image of B₀ under f. - Formally: pre0image f^-1(B₀) = {a|f(a)∈B₀}.

m-tuple

- Let m be a positive integer. Given a set X, we define an m-tuple of elements of X to be a function x:{1,...,m}→X. - If x is an m-tuple, we often denote the value of x at i by the symbol xi, rather than x(i), and call it the ith coordinate of x.

Cartesian product of J-tuples

- Let {Aα}α∈J be an indexed family of sets. Let X=∪Aα. The cartesian product of this indexed family, denoted by ΠAα is defined to be the set of all J-tuples of elements of X such that x_α∈Aα for each α∈J.

Box topology

- Let {Xα}α∈J be an indexed family of topological spaces. Let us take as a basis for a topology on the product space ΠXα the collection of all sets of the form ΠUα, where Uα is open in Xα. The topology generated by this basis is called the box topology.

Indexing function, index, indexed family of sets

- Let å be a nonempty collection of sets. An indexing function for å is a surjective function f from some set J, called the index set, to å. The collection å, together with the indexing function f, is called an indexed family of sets. Given a∈J, we shall denote the set f(α) by the symbol Aα. - Denote the indexed family by: {Aα}α∈J.

Projections

- Let π₁:XxY→X be defined by the equation π₁(x,y)=x. - Let π₂:XxY→Y be defined by the equation π₂(x,y)=y. - The maps above are the projections of XxY onto its first and second factors (onto because its surjective).

Power set of A

- Set of all subsets of A.

Let X be a topological space. Suppose that C is a collection of open sets of X such that for each open set U of X, and each x in U, there is an element c of C such that

- Such that x∈c⊂U. - Then C is a basis for the topology of X.

Largest element, smallest element

- Suppose A is a set order by the relation <. Let A₀ be a subset of A. We say that b is the largest element of A₀ if b∈A₀ and if x≤b for every x∈A₀. - a is the smallest element of A₀ if a∈A₀ and if a≤x for every x∈A₀.

Imbedding

- Suppose f:X→Y is an injective continuous map, where X and Y are top spaces. Let Z be the image set f(X), a subspace of Y. Then the function f':X→Z obtained by restricting the range of f is bijective. - If f' happens to be a homeomorphism of X with Z, then the map f:X→Y is an imbedding of X in Y.

Order type

- Suppose that A and B are two sets with order relations <_a (<₀) and <_b (<₀₀). We say that A and B hace the same order type if there is a bijective correspondence between them that preserves order; that is, if there exists a bijective function f:A→B such that a₁<₀a₂ → f(a₁)<₀₀f(a₂).

Dictionary order relation on AxB

- Suppose that A and B are two sets with order relations <_a and <_b. Define an order relation < on AxB by defining a₁xb₁<a₂xb₂ if a₁(<_a)a₂ or if a₁=a₂ and b₁(<_b)b₂.

Finer, coarser, comparable

- Suppose that T and T' are two topologies on a given set X. If T'⊃T, we say that T' is finer than T, and T is coarser than T'. - If T' properly contains T, we say that T' is strictly finer than T, and T is strictly coarser than T'. - We say T is comparable with T' if either T'⊃T or T⊃T'.

Let Y be a subspace of X, and A a subset of Y. A° is the closure of A in X. Then

- The closure of A in Y equals A°∩Y.

Closure

- The closure of A is defined as the intersection of all closed sets containing A. - Denoted A bar (or A°), and it is a closed set. - A is closed, A=A°.

Let A be a subset of the topological space X. Let A' be the set of all limit points of A. Then

- Then A°=A∪A'.

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.

Let f:A→B. If there are functions g:B→A and h:B→A such that g(f(a))=a for every a in A and f(h(b))=b for every b in B,

- Then f is bijective and g=h=f^-1.

Let X denote the two element set {0,1}.

- Then the set X^w is uncountable.

A is a proper subset of B

- There is at least one element of B that is not in A.

Bounded above, upper bound, least upper bound/supremum

- We say that the subset A₀ of A is bounded above if there is an element b of A such that x≤b for every x∈A₀. - The element b is called an upper bound for A₀. - If the set of all upper bounds for A₀ has a smallest element, that element is called the least upper bound/supremum of A (denoted supA₀).

Bounded below, lower bound, greatest lower bound/infimum

- We say that the subset A₀ of A is bounded below if there is an element a of A such that a≤x for every x∈A₀. - The element a is called a lower bound for A₀. - If the set of all lower bounds for A₀ has a largest element, that element is called the greatest lower bound/infimum of A (denoted infA₀).