Topology Exam 1

Theorem 5.6

ANY collection of subsets on a set X is a subbase

Equivalence Relation

A reflexive, symmetric, and transitive relationship

Antisymmetric

A relation R on A is antisymmetric iff aRb and bRa implies a = b ex. ≤

Reflexive

A relation R on A is reflexive iff aRa for each a ∈ A

Pseudometric

A set M and a function p (M, p) that satisfies all requirements of a metric space M.a) through M.d), except for the second part of M.b). It does not satisfy the requirement that p(x, y) = 0 implies x = y

Boundary point

A point x such that for any ε > 0, U(x, ε) contains a point in S and a point not in S. For every ε-disk around the boundary point, the disk is partially in S and partially not in S.

Interior point

A point x such that there exists an ε > 0 such that U(x, ε) is a subset of the set S. There is an ε-disk around our point contained completely in S.

Exterior point

A point x such that there exists an ε > 0 such that U(x, ε) ∩ S ≠ ∅. There is an ε-disk around the point that is not contained in S at all.

Symmetric

A relation R on A is symmetric iff aRb implies bRa for each a,b ∈ A

Transitive

A relation R on A is transitive iff aRb and bRc implies aRc for each a,b, c ∈ A

Partial Order

A relation R that is reflexive, antisymmetric, and transitive

Strict Order

A relation R that is transitive and NOT symmetric and NOT reflexive ex. <

Denumerable

A set A is denumerable iff A is equipotent with N |A| = ℵ This means it is countable; we can match the set in a one-to-one matching with the natural numbers

Cardinal of the Continuum

A set A is said to have the cardinal of the continuum iff A is equipotent with R |A| = C

Well-Ordered

A set A is well-ordered if it has a linear order such that every subset of A has a smallest element

Open

A set E in (M, p) is open iff for each x ∈ E, there is an ε-disk about x contained in E. We can find at least one ε-disk around every x in E, where the ε-disk is contained completely in E. To show a set is not open, find an x with NO ε-disk around it contained in E.

Closed

A set is closed iff it is the complement of an open set.

Radial Topology

Call a subset of R² radially open iff it contains an open line segment in each direction around each of its points.

DeMorgan's Laws

(∪Sα)c = ∩(Sαc) ***The complement of the union is the intersection of all the complements (∩Sα)c = ∪(Sαc) ***The complement of the intersections is the union of all the complements.

Theorem 5.3

"B" is a base for a topology iff a. X = ∪ B ***through unions of the elements in our base, we can obtain the whole space b. whenever B₁, B₂ ∈ "B" with p ∈ B₁ ∩ B₂, there is some B₃ ∈ "B" with p ∈ B₃ ⊂ B₁ ∩ B₂ ***every time we intersect two elements of the basis, this intersection must already be in there

Closurey

Cl(E) = ∩ {K ⊂ X | K is closed and E ⊂ K} The intersection of all closed sets K such that E ⊂ K E plus its boundary points; E plus whatever makes it closed

Topological definition of Continuity

A function f(x) is continuous when the inverse image of an open set is open. Again, we first pick our open interval (the outputs, y-axis). Then, we look at what maps to this interval (the inputs, x-axis). If this set that maps to our interval is open, we have a continuous function. But, if this set is not open, then we are discontinuous.

Nhood base

A nhood base at x in a topological space X is a subcollection, Bx, taken from the nhood system Ux having the property that each U ∈ Ux contains some V ∈ Bx Ux = {U ⊂ X | V ⊂ U for some V ∈ Bx} For every nhood we pick around x, there is an element of the nhood base Bx, contained within it.

Lattice

A partially ordered set L is a lattice iff for each two element set {a, b} in L has a least upper bound (a v b) and a greatest lower bound (a ∧ b). In our lattice of topologies, we always have a weakest topology (the trivial topology) and a strongest topology (the discrete)

Linearly Ordered

A set is linearly ordered by a partial order provided that for any a, b ∈ A, exactly one of the following hold: a < b, b < a, a = b If we can put our elements in a line from smallest to largest, then we have a linear order. Our relation stronger/weaker was NOT a linear order: there were some topologies that were neither weaker nor stronger than the other

Clopen

A set that is both closed and open. A set and its compliment are both open. ***The space X and ∅ are always clopen.

Neither

A set that is not open or closed. A set where it is not open and its compliment is not open.

Metric Space

An ordered pair (M, p) satisfying: M.a) p(x, y) ≥ 0 M.b) p(x, x) = 0; p(x, y) = 0 implies x = y M.c) p(x, y) = p(y, x) M.d) p(x, y) + p(y, z) ≥ p(x, z) ...a method of measuring distance. The functions p are the distance functions

Topology

Collection of open subsets satisfying: G.1) Any union of elements of T belongs to T ***Closed under arbitrary unions G.2) Any finite intersection of elements of T belongs to T ***Closed under finite intersections G.3) ∅ and X belong to T ↑This is how we define a topology using open sets: these open sets, that satisfy these conditions, are defined to us

Theorem 3.4 (defining a topology by closed sets)

F is the collection of closed sets in a topological space: F.a) Any intersection of elements belongs to F ***Closed under arbitrary intersections F.b) Any finite union of elements belongs to F ***Closed under finite unions F.c) X and ∅ belong to F Conversely, given a set X and any family F of subsets satisfying F.a), F.b), and F.c), the collection of complements of members of F is a topology on X; its family of closed sets is F ↑How we define a topology using closed sets. We have a group of sets that satisfies the above conditions. Thus, their complements are the open sets and members of our topology.

Frontier

Fr(E) = Cl(E) ∩ Cl(X - E) These are our boundary points.

Weaker versus Stronger

Given two Topologies, T1 and T2, T1 ⊂ T2 (allows for T1 = T2) T1 is weaker/stronger/coarser T2 is stronger/finer/larger ***To prove: If a set is open in T1, then it will be open in T2. ***To prove they are equal: do from both directions.

Metric Space definition of Continuity

If (M, p) and (N, o) are metric spaces, a function f: M → N is continuous at x in M iff for each ε > 0, there is some δ > 0 such that o(f(x), f(y)) < ε whenever p(x, y) < δ

Base for a topology

If (X, T) is a topological space, a base for T is a collection B ⊂ T such that T = {U B, B ∈ C | C ⊂ B) Our base is composed of elements from our topology (the elements in the base are all open). What elements do we need from T, where, when we do all the possible unions we end up with T?

Subbase

If (X, T) is a topological space, a subbase for T is a collection C ⊂ T such that the collection of all finite intersections of elements from C forms a basis for T ***We have a set; when we add intersections of this set, we obtain a basis for our T. Note that empty intersections leads to our whole space. We take ALL unions of this basis and get the open sets of our topology

Neighborhood

If X is a topological space and x ∈ X, a neighborhood (nhood) of x is a set U which contains an open set V containing x. The neighborhood need not be open. It only must contain and open set from our topology. x must be a member of this open set.

Greatest lower bound

In topologies: the largest topology that was still a subset of two given topologies a ∧ b

Least upper bound

In topologies: the smallest topology that contains both of the two given topologies a v b

Interior

Int(E) = ∪ {G ⊂ X | G is open and G ⊂ E} The union of all open sets G such that G ⊂ E

ε-disk

Let (M, p) b a metric space, let x be a point of M. For ε > 0, we define Up(x, ε) = {y ∈ M | p(x, y) < ε} All points that are within an ε distance away from the given x.

Calculus definition of Continuity

Let f(x) be a function defined on an interval that contains x = a. Then we say f(x) is continuous at x = a if for every number ε > 0 there is some δ > 0 such that |f(x) - f(a)| < ε whenever 0 < |x - a| < δ ***We set our ε bounds around f(a). Then, we can find a δ around a such that f(a - δ) to f(a + δ) gives us values that are in the ε bounds we set.

Theorem 4.5 (defining a topology by specifying a base)

Let x be a topological space and for each x ∈ X, let Bx be a nhood base at x. Then V.a) If V ∈ Bx, then x ∈ V V.b) If V₁, V₂ ∈ Bx, then there is some V₃ ∈ Bx such that V₃ ⊂ V₁ ∩ V₂ V.c) If V ∈ Bx, there is some V₀ ∈ Bx such that if y ∈ V₀, then there is some W ∈ By with W ⊂ V V.d) G ⊂ X is open iff G contains a basic nhood at each of its points Conversely, in a set X, if a collection Bx of subsets of X is assigned to each x ∈ X so as to satisfy V.a), V.b), and V.c), and if we define "open" using V.d), the result is a topology on X in which Bx is a nhood base at x, for each x ∈ X ↑How we define a topology by specifying a base.

Lower-bound Topology

On the real line R: A subset S of R is open iff for every x in S, the interval [a, b) is contained in S; x ∈ [a, b)

Discrete Topology

Our collection of open sets is the power set.

Trivial Topology

T = {∅, X}

Nhood system

The collection Ux of all nhoods of x

Discrete Metric

The distance between any two points is 1; the distance between a point and itself it 0. The discrete metric views all sets as both open and closed: if ε > 1, then our open set is X; if 0 ≤ ε < 1, then the open set is just the point. ***We have no boundary points.

Theorem 3.11 (defining a topology with an interior operation)

The interior operation A → Int(A) in a topological space X has the following properties. I.a) Int(A) ⊂ A I.b) Int(Int(A)) = Int(A) I.c) Int(A ∩ B) = Int(A) ∩ Int(B) I.d) Int(X) = X I.e) G is open iff Int(G) = G Conversely, given any map A → Int(A) of P(X) into P(X) in a set X, satisfying I.a) through I.d), if open sets are defined in X using I.e), the result is a topology on X in which the interior of a set A ⊂ X is just Int(A) ↑How we define a topology using an interior operation. When we meet I.a) through I.d), we use I.e) to define our open sets. These open sets are those of our topology.

Usual Topology

The metric topology created my our usual metric on any subset of R²

Theorem 3.7 (defining topology with a closure operation)

The operation A → Cl(A) in a topological space has the following properties: K.a) E ⊂ Cl(E) K.b) Cl(Cl(E)) = Cl(E) ***When we close an already closed set, nothing changes K.c) Cl(A ∪ B) = Cl(A) ∪ Cl(B) K.d) Cl(∅) = ∅ K.e) E is closed in X iff Cl(E) = E Moreover, given a set X and a mapping A → Cl(A) of P(X) into P(X) satisfying K.a) through K.d), if we define closed sets in X using K.e), the result is a topology on X whose closure operation is just the operation A → Cl(A) we began with. ↑How we define a topology using a closure operation. When we meet K.a) through K.d), we use K.e) to define closed sets. Once we know our closed sets, we can define open sets of our topology.

Maximum Metric

The space R² with p((x₁, x₂), (y₁, y₂)) = max{ |x₁ - y₁|, |x₂ - y₂| }

Taxicab Metric

The space R² with p((x₁, x₂), (y₁, y₂)) = |x₁ - y₁| + |x₂ - y₂|

Usual Metric

The space R² with p((x₁, x₂), (y₁, y₂)) = √((x₁ - y₁)² + (x₂ - y₂)²) This is our usual distance function