Intermediate Analysis

Question 1

Neighborhood

Answer 1

N(x;e) = { y within Reals: |x-y| < e } where e is the radius

Question 2

N(x;e) = { y within Reals: |x-y| < e } where e is the radius

Answer 2

Neighborhood

Question 3

Deleted Neighborhood

Answer 3

N*(x;e) = N(x,e) \ {x}

Question 4

N*(x;e) = N(x,e) \ {x}

Answer 4

Deleted Neighborhood

Question 5

Interior Point

Answer 5

There exist epsilon greater than 0 such that the neighborhood of x is the subset of set (S)

Question 6

There exist epsilon greater than 0 such that the neighborhood of x is the subset of set (S)

Answer 6

Interior Point

Question 7

Boundary Point

Answer 7

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)

Question 8

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)

Answer 8

Boundary Point

Question 9

Accumulation Point

Answer 9

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

Question 10

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

Answer 10

Accumulation Point

Question 11

isolated point

Answer 11

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}

Question 12

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}

Answer 12

isolated point

Question 13

compact set

Answer 13

A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.

Question 14

A subset S of a topological space X is —– if for every open cover of S there exists a finite subcover of S.

Answer 14

compact set

Question 15

Subcover

Answer 15

Let C be a cover of a topological space X. A subcover of C is a subset of C that still covers X

Question 16

Let C be a cover of a topological space X. A —- of C is a subset of C that still covers X

Answer 16

Subcover

Question 17

Open cover

Answer 17

A collection of open sets of a topological space whose union contains a given subset.

Question 18

A collection of open sets of a topological space whose union contains a given subset.

Answer 18

Open Cover

Question 19

Cover

Answer 19

a cover of a set X is a collection of sets whose union contains X as a subset

Question 20

a —– of a set X is a collection of sets whose union contains X as a subset

Answer 20

Cover

Question 21

First Archimedean Prop

Answer 21

For every x within naturals, there exists n within reals such that x is less than n

Question 22

For every x within naturals, there exists n within reals such that x is less than n

Answer 22

First Archimedean Prop

Question 23

Second Archimedean Prop

Answer 23

For every x greater than 0. There exist n within naturals such that 0 is less than 1/n less than x

Question 24

For every x greater than 0. There exist n within naturals such that 0 is less than 1/n less than x

Answer 24

Second Archimedean Prop

Question 25

Third Archimedean Prop

Answer 25

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

Question 26

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

Answer 26

Third Archimedean Prop

Question 27

Open Set

Answer 27

For every x within set (s) there exist epsilon greater than 0 such that the neighborhood of x is a subset of set (s)

Question 28

For every x within set (s) there exist epsilon greater than 0 such that the neighborhood of x is a subset of set (s)

Answer 28

Open Set

Question 29

Closed

Answer 29

if complement of the set is open or it contains all of the boundary points

Question 30

if complement of the set is open or it contains all of the boundary points

Answer 30

Closed