Chapter 1&2 Flashcards
Topological space
The set X together with a topology T on X
Trivial topology
T={empty set, X}
Discrete Topology
T= collection of all subsets of X
Finite Complement Topology
T = empty set and every set in R with a finite complement
Finer
Coarser
If T_1 is a subset of T_2 then T_2 is ______ than T_1 and T_1 is _____ than T_2
A topology T on A set X is…
A collection of subsets of X, (open sets) such that:
1) empty set and X are open sets
2) the intersection of finitely many open sets is an open set
3) the Union of any collection of open sets is an open set
Neighborhood of x
Let X be a topological space and x in X. An open set U containing x is said to be a…
Thm 1.4
Let X be a topological space and let A be a subset of X. Then A is open if and only if for each x in A, there is a neighborhood U of x such that x in U is a subset of A.
Basis for a topology on X
1) for each x in X there is a B in ‘B’ such that x in B
2) if B_1 and B_2 are in ‘B’ and x in B_1 intersect B_2, then there exists B_3 in ‘B’ such that x in B_3 subset B_1 intersect B_2
The topology T generated by a basis, ‘B’, on a set X…
Is generated by defining the open sets to be the empty set and every set that is equal to a union of basis elements
Standard topology
Topology generated by ‘B’ ={(a,b) subset of R | a<b></b>
Let ‘B’ be a basis. Assume that B_1,…,B_n in ‘B’ and that x in intersection _i=1 ^n B_i. Then…
There exists B’ in ‘B’ such that x in ‘B’ subset intersection _i=1 ^n B_i
Lower limit topology
Generated by ‘B’={[a,b) subset R|a<b></b>
Upper limit topology
Generated by ‘B’ ={(a,b] subset R| a<b></b>
Digital line topology
B(n) = {n} if n is odd
{n-1, n, n+1} if n is even
Generated by
‘B’={B(n)| n in Z}
Let X be a set and ‘B’ be a basis for a topology on X. Then U is open in the topology generated by ‘B’ if and only if
For each x in U there exists a basis element B_x in ‘B’ such that x in B_x subset U
Basis for a topology on R^2
Collection ‘B’={B(x, e)|x in R^2, e>0}
Let y be in R^2 and assume r>0. Then for every x in B(y,r)…
There exists an e>0 such that B(x, e) subset B(y, r)
Basis for standard topology on R^2
‘B’={(a,b)x(c,d) subset R^2|a<b></b>
Let X be a set with topology T, and let C be a collection of open sets in X. If, for each open set U in X and for each x in U, there is
A basis that generates the topology T
Vertical line topology
Topology generated by ‘B’={{a}x(b,c) subset R^2 | a,b,c in R}
Closed
Subset A of a topological space X is ____ if the set X-A is open
The closed ball of radius e centered at x
For each x in R^2 and e>0. B(x,e)={y in R^2| d(x,y) less than or equal to e}
Closed rectangle
If [a,b] and [c,d] are closed bounded intervals in R, then the product [a,b]x[c,d] subset R^2 is called
D(x,y)
Euclidean distance between x and y
Closed balls and closed rectangles are…
Closed sets in the standard topology on R^2
Let X be a topological space. The following statements about collection of closed sets in X hold:
1) empty set and X are closed
2) the intersection of any collection of closed sets is a closed set
3) the Union of finitely many closed sets is a closed set
Hausdorff
If for every pair of distinct points x and y in X, there exists disjoint neighborhoods U and V of x and y respectively
If X is a Hausdorff space, then….
Every single-point subset of X is closed
Infinite comb
Subset of the plane defined by C={(x,0)|0 leq x leq 1} U {(1/2^n,y)| n=0,1,2,… and 0 leq y leq 1}
Let A be a subset of topological space X. The interior of A is
The union of all open sets contained in A.
Let A be a subset of a topological space X. The closure of A
Is the intersection of all closed sets containing A
Let X be a topological space and A and B be subsets of X.
If U is an open set in X and U subset A
Then U subset Int(A)
Let X be a topological space and A and B be subsets of X.
If C is a closed set in X and A subset C
Then Cl(A) subset C
Let X be a topological space and A and B be subsets of X.
If A subset B then
Int(A) subset Int(A)
Let X be a topological space and A and B be subsets of X.
If A subset B then
Cl(A) subset Cl(B)
Let X be a topological space and A and B be subsets of X.
A is open if and only if
A=Int(A)
Let X be a topological space and A and B be subsets of X.
A is closed if and only if
A=Cl(A)
Dense
If Cl(A) = X for a subset of a topological space X then it is called
Let X be a topological space, A be a subset of X, and y be an element of X. Then y in Int(A) if and only if
There exists an open set U such that y in U subset A
Let X be a topological space, A be a subset of X, and y be an element of X. Then y in Cl(A) if and only if
Every open set containing y intersects A
For sets A and B in a topological space X, the following statements hold:
1) Int(X-A)=X - Cl(A)
2) Cl(X-A)= X - Int(A)
3) Int(A)UInt(B) subset Int(AUB)
4) Int(A) intersect Int(B) = Int(A intersect B)
Limit point
Let A be a subset of a topological space X. A point x in X is a ____ of A if every neighborhood of x intersects A in a point other than x.
Let A be a subset of a topological space X, and let A’ be the set of limit points of A. Then…
Cl(A)=A U A’
A subset A of a topological space is closed if and only if
It contains all of its limit points
In a topological space X, a sequence (x_1,x_2,…) converges to x in C if
For every neighborhood U of x, there is a positive integer N such that x_n in U for all n geq N.
Let A be a subset of R^n in the standard topology. If x is a limit point of A, then…
There is a sequence of points in A that converges to x
If X is a Hausdorff space then
Every convergent sequence of points in X converges to a unique point in X
Let A be a subset of a topological space X. The boundary of A is
Cl(A)-Int(A)
Let A be a subset of a topological space X and let x be a point in X. Then x in boundary of A if and only if
Every neighborhood of x intersects both A and X-A
Let A be a subset of a topological space X. Then the following statements about the boundary of A hold:
1) boundary of A is closed
2) boundary of A = Cl(A) intersect Cl(X-A)
3) boundary of A intersect Int(A) = empty set
4) boundary of A U Int(A) = Cl(A)
5) boundary of A subset A if and only if A is open
6) boundary A intersect A = empty set if and only if A is open
7) boundary of A = empty set if and only if A is both open and closed
