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
