Intermediate Analysis Flashcards
Neighborhood
N(x;e) = { y within Reals: |x-y| < e } where e is the radius
N(x;e) = { y within Reals: |x-y| < e } where e is the radius
Neighborhood
Deleted Neighborhood
N*(x;e) = N(x,e) \ {x}
N*(x;e) = N(x,e) \ {x}
Deleted Neighborhood
Interior Point
There exist epsilon greater than 0 such that the neighborhood of x is the subset of set (S)
There exist epsilon greater than 0 such that the neighborhood of x is the subset of set (S)
Interior Point
Boundary Point
Fall all epsilon greater than 0, there exist y within the neighborhood of x and intersection of set (S) also there exist z within the neighborhood of x and intersection of the complement of set (S)
Fall all epsilon greater than 0, there exist y within the neighborhood of x and intersection of set (S) also there exist z within the neighborhood of x and intersection of the complement of set (S)
Boundary Point
Accumulation Point
is a point x that can be “approximated” by points of S in the sense that every neighborhood of x with respect to the topology on X also contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S
is a point x that can be “approximated” by points of S in the sense that every neighborhood of x with respect to the topology on X also contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S
Accumulation Point
isolated point
There exist epsilon greater than 0 such that the neighborhood of x and intersection of set (S) equal the set that contains only the element {x}
There exist epsilon greater than 0 such that the neighborhood of x and intersection of set (S) equal the set that contains only the element {x}
isolated point
compact set
A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.
A subset S of a topological space X is —– if for every open cover of S there exists a finite subcover of S.
compact set
Subcover
Let C be a cover of a topological space X. A subcover of C is a subset of C that still covers X
Let C be a cover of a topological space X. A —- of C is a subset of C that still covers X
Subcover
Open cover
A collection of open sets of a topological space whose union contains a given subset.
A collection of open sets of a topological space whose union contains a given subset.
Open Cover
Cover
a cover of a set X is a collection of sets whose union contains X as a subset
a —– of a set X is a collection of sets whose union contains X as a subset
Cover
First Archimedean Prop
For every x within naturals, there exists n within reals such that x is less than n
For every x within naturals, there exists n within reals such that x is less than n
First Archimedean Prop
Second Archimedean Prop
For every x greater than 0. There exist n within naturals such that 0 is less than 1/n less than x
For every x greater than 0. There exist n within naturals such that 0 is less than 1/n less than x
Second Archimedean Prop
Third Archimedean Prop
For every x greater than 0 and for all y within the reals, there exists n within naturals such that n times x is greater than y
For every x greater than 0 and for all y within the reals, there exists n within naturals such that n times x is greater than y
Third Archimedean Prop
Open Set
For every x within set (s) there exist epsilon greater than 0 such that the neighborhood of x is a subset of set (s)
For every x within set (s) there exist epsilon greater than 0 such that the neighborhood of x is a subset of set (s)
Open Set
Closed
if complement of the set is open or it contains all of the boundary points
if complement of the set is open or it contains all of the boundary points
Closed
