Topology Midterm

Question 1

countable (defn, ex, ex, Prop)

Answer 1

A set X is countable if X is bijective to N OR if X is finite

ex. R is not countable
ex. NxN is countable

Prop: countable union of countable sets is countable

Question 2

algebraic, transcendental, Theorem

Answer 2

A number is called algebraic if it is a solution to some integer polynomial. If a number is not algebraic then it is called transcendental

Theorem: the set of algebraic numbers is countable

Question 3

metric space

Answer 3

We say (X,d) is a metric space if it has a metric d with

• d(x,y)=0 <=> x=y
• d(x,y) ≤ d(x,z) + d(z,y)
• d(x,y) = d(y,x)

Question 4

limit (metric space)

Answer 4

If {x_n} is a sequence of points in X, we say lim x_n = x if for every epsilon>0 there exists N such that for all n≥N, d(x_n,x)

Question 5

continuous (metric space)

Answer 5

If X and Y are metric spaces, we say f:X->Y is continuous if whenever lim x_n=x then lim f(x_n)=f(x)

Question 6

homeomorphism

Answer 6

f:X->Y is called a homeomorphism if f is bijective, continuous and f^{-1} is continuous

Question 7

Topological space

Answer 7

A topological space is a set X equipped with a collection of subsets (called open subsets) such that

• ø,X are open
• arbitrary unions of open sets are open
• finite intersections of open sets is open

ex. T={ø,X} is called the trivial topology
ex. T=P(X) is called the discrete topology

Question 8

coarser/finer

Answer 8

If T_1 subset T_2 then we say T_2 is finer and T_1 is coarser

Question 9

continuous

Answer 9

We call a map f:X->Y between topological spaces continuous if for all x in X, for all open U in Y, f(x) in U then there exists an open x in V \subset X such that f(V) subset U

Equivalently, f is continuous if f^{-1}(open) is open

Question 10

basis

Answer 10

A collection B of subsets of X is called a basis for the topology if

• for every x in X, there exists some B such that x in B
• If B1,B2 are in the basis and x in B_1 \cap B_2 then there exists some B_3 such that x in B_3 \subset B_1 cap B_2

Question 11

limit point

Answer 11

A point x is called a limit point of A if every open subset U of x contains a point in A

Question 12

Hausdorff (defn + Theorem)

Answer 12

A topological space is called Hausdorff if for every x≠y there exists disjoint open subsets U and V with x in U, y in V

Theorem: If X is Hausdorff then whenever a sequence in X has a limit, the limit is unique

Question 13

Quotient space

Answer 13

Let X be a topological space, ~ an equivalence relation on X, so

π: X -> X/~
x -> [x]

where a subset V subset X/~ is open <=> π^-1(V) is open in X

Question 14

product topology

Answer 14

The product topology on ∏X_alpha is generated by the basis ∏U_alpha where each U_alpha is open in X_alpha and U_alpha=X_alpha for all but finitely many alpha

Question 15

separation / connected, Prop

Answer 15

A separation of X is a disjoint pair U and V such that U and V are open, X = UuV.

X is connected if there is no separation.

Prop: X is connected <=> the only clopen (open and closed) subsets of X are X and itself

Theorem: If X=UX_alpha such that each X_alpha is connected and intersection of X_alpha is non-empty

Question 16

path-connected

Answer 16

If X is a topological space, for every x,y in X there exists a continuous path f:[0,1]->X continuous such that f(0)=x, f(1)=y

Question 17

compact + Theorem + Theorem + Theorem

Answer 17

A topological space X is compact if any open covering of X has a finite subcovering

Theorem: Any closed subspace of a compact space is compact

Theorem: Any compact space of a Hausdorff space is closed

Theorem: The image of a compact space under a continuous map is again compact

Question 18

finite intersection property + Theorem

Answer 18

A collection C of subsets in X has finite intersection property if for any finite subcollection, the intersection is non-empty

Theorem: X is compact <=> every collection of closed subsets in X which has the finite intersection property has non-zero intersection

Question 19

limit point compactness + Theorem

Answer 19

X is limit point compact if every infinite subset has a limit point

Theorem: compactness => limit point compact

Question 20

Net

Answer 20

A net is some {x_\alpha} where J is a directed set

• J has a partial order
• – for all alpha, alpha ≤ alpha
• – alpha ≤ beta, beta ≤ alpha => alpha = beta
• – alpha ≤ beta, beta ≤ gamma => alpha ≤ gamma
• Given an alpha, beta in J, there exists some gamma such that gamma ≥ alpha, gamma ≥ beta

Proposition: If p is a limit point of A, there exists a net in A such that lim x_alpha = x

Question 21

locally compact

Answer 21

X is locally compact if for every p there exists a neighborhood U of p which is contained in a compact subspace of X

Question 22

compactification

Answer 22

Let X and Y be topological spaces. Y is called the compactification of X if

• X subset Y
• Y is compact
• X is dense in Y

Question 23

one point compactification

Answer 23

Let X be a locally compact topological space, Y = XU{∞}. Define a basis for the topology on Y by

• open subsets of X
• Y\C where C is any compact subspace of X

We call Y the one-point compactification

Question 24

first/second countable

Answer 24

X is first countable if there exists a countable basis at x

X is second countable if it has a countable basis

Question 25

regular + Theorem + Theorem

Answer 25

Suppose every singleton in X is closed. X is called regular if for every x and closed subset C of X with x not in C, there exists disjoint open U,V such that x in U, C subset V

Theorem: metric space is regular

Theorem: A compact Hausdorff space is regular

Question 26

normal + Theorem

Answer 26

Suppose every singleton in X is closed. X is called normal if for every disjoint closed subsets A,B then there exists disjoint open subsets separating A and B

Theorem: A metric space is normal

Question 27

embedding

Answer 27

A continuous map f:X->R^n is called an embedding if

• f is injective
• f:X -> f(X) is a homeomorphism where f(X) is given the subspace topology of R^n

Question 28

m-manifold + Theorem

Answer 28

An m-manifold is a second countable topological space such that each point in M has a neighborhood homeomorphic to R^m

Theorem: If M is a compact m-manifold, then there exists an embedding f:M -> R^n for some n

Question 29

Theorem (partitioning of unity)

Answer 29

Theorem: Let X be a normal topological space. If {U_i}_{i=1}^k is a finite open cover then there exists continuous functions {phi_i} on X such that

• phi_i : X -> [0,1] with supp(phi_i)\subset U_i
• phi_i(x)+phi_2(x)+…+phi_k(x)=1

We call {phi_i} a partitioning of unity subordinate to {U_i}

Question 30

refinement

Answer 30

Given two open coverings of X, \A and \B, we say \B is a refinement on \A if for every B in \B there exists an A in \A such that B subset A

Question 31

order m+1

Answer 31

A collection \A of subsets of X is said to be of order m+1 if some point of X lies in m+1 elments of \A and no point of X lies in more than m+1 elements of \A

Question 32

finite dimension / topological dimension + Theorem + Theorem

Answer 32

We say X is of finite dimension if there exists m such that every open cover \A of X has a refinement \B of order at most m+1

The topological dimension of X is the smallest m for which this holds and we write dim(X)=m

Theorem: If dim(X)

Question 33

complete

Answer 33

A metric space is called complete if every Cauchy sequence {x_n} has a limit point in X

Question 34

completion + Theorem

Answer 34

If (Y,d_Y) is a metric space, the completion of (Y,d_Y) is a complete metric space (X,d_X) such that there exists an inclusion i:(Y,d_Y)->(X,d_X) satisfying

• d_X(i(y),i(z)) = d_Y(y,z)
• X=overline{Y}

Theorem: completion always exists

Question 35

contraction + Theorem

Answer 35

If (X,d) is a complete metric space, then f:(X,d)->(X,d) is a contraction if there exists 0