Topology Basics

Question 1

What are intervals in the context of the ordered field structure of R?

Answer 1

Intervals are special subsets of R.

Question 2

What is the definition of an open interval (a, b)?

Answer 2

The open interval is the set (a, b) := {x ∈ R : a < x < b}.

Question 3

What is the definition of a closed interval [a, b]?

Answer 3

The closed interval is the set [a, b] := {x ∈ R : a ≤ x ≤ b}.

Question 4

What is the definition of a half-open (or half-closed) interval?

Answer 4

The half-open intervals are defined as (a, b) := {x ∈ R : a < x < b} and [a, b) := {x ∈ R : a ≤ x < b}.

Question 5

What are the endpoints of an interval?

Answer 5

The points a and b are called the endpoints of the interval.

Question 6

What is the length of an interval?

Answer 6

Each of the intervals is bounded and has length defined by b - a.

Question 7

What happens if a = b in terms of intervals?

Answer 7

If a = b, the corresponding open interval is the empty set (a, a) = ∅, and the corresponding closed interval is the singleton set [a, a] = {a}.

Question 8

How are unbounded intervals defined?

Answer 8

Unbounded intervals can be defined by replacing one or both endpoints with the symbols +∞ and -∞.

Question 9

What are the definitions of infinite open intervals?

Answer 9

The infinite open intervals are defined as (a, ∞) := {x ∈ R : x > a} and (-∞, b) := {x ∈ R : x < b}.

Question 10

What are the definitions of infinite closed intervals?

Answer 10

The infinite closed intervals are defined as [a, ∞) := {x ∈ R : x ≥ a} and (-∞, b] := {x ∈ R : x ≤ b}.

Question 11

What is the definition of the infinite interval (-∞, ∞)?

Answer 11

The infinite interval is the entire set R, i.e., (-∞, ∞) := R.

Question 12

What is the definition of an 𝜖-neighbourhood of a point c?

Answer 12

The set of all points in the open interval (c - 𝜖, c + 𝜖)

Denoted as N𝜖(c) for every 𝜖 > 0.

Question 13

What does it mean for a set S to be a neighbourhood of a point c?

Answer 13

There exists an open interval (a, b) such that c ∈ (a, b) ⊆ S

This means any open interval containing c must be a neighbourhood of c.

Question 14

Is the closed interval [1, 2] a neighbourhood of the point 1?

Answer 14

No

Although it contains the point 1, it is not an open interval.

Question 15

What is the result of the union of two neighbourhoods of a point?

Answer 15

The union is also a neighbourhood of the point

If S1 and S2 are neighbourhoods of c, then S1 ∪ S2 is a neighbourhood of c.

Question 16

True or False: The intersection of two neighbourhoods of a point is also a neighbourhood of the point.

Answer 16

True

If S1 and S2 are neighbourhoods of c, then S1 ∩ S2 is a neighbourhood of c.

Question 17

What happens when you take the union of a finite number of neighbourhoods of a point?

Answer 17

It is also a neighbourhood of the point

This follows from the conclusion about the union of two neighbourhoods.

Question 18

What happens when you take the intersection of a finite number of neighbourhoods of a point?

Answer 18

It is also a neighbourhood of the point

This is based on the conclusion regarding the intersection of two neighbourhoods.

Question 19

What is the result of the union of an infinite number of neighbourhoods of a point?

Answer 19

It is also a neighbourhood of the point

If Sα where α ∈ A are neighbourhoods of c, then their union is a neighbourhood of c.

Question 20

Can the intersection of an infinite number of neighbourhoods of a point be a neighbourhood of that point?

Answer 20

No, it may not be

For example, the intersection of intervals that shrink down to a point may not contain any open intervals.

Question 21

Fill in the blank: A set S is a neighbourhood of a point c if and only if there exists a natural number n such that _______.

Answer 21

(c - n, c + n) ⊆ S

Question 22

What is the example of a 0.001-neighbourhood of the point 5?

Answer 22

(4.999, 5.001)

This is derived from the definition of 𝜖-neighbourhood.

Question 23

What is an interior point of a set S?

Answer 23

A point z in S is called an interior point of S if a neighbourhood N(¢) of z is contained in S.

This means N(z) ∩ S = N(z).

Question 24

What is the notation for the set of all interior points of S?

Answer 24

int S, Sᵢ, or S°

The interior of S is always a subset of S.

Question 25

What defines an exterior point of a set S?

Answer 25

A point z is an exterior point of S if it belongs to the interior of the complement of S.

This means N(z) ∩ S = ∅.

Question 26

What is the notation for the set of all exterior points of S?

Answer 26

ext S or Sᵉ

The exterior of S is always a subset of the complement of S.

Question 27

How is a boundary point of S defined?

Answer 27

A point z is a boundary point of S if it is neither an interior point nor an exterior point of S.

For every neighbourhood N(z), both N(z) ∩ S and N(z) ∩ S° are non-empty.

Question 28

What is the notation for the set of all boundary points of S?

Answer 28

∂S

This notation represents the boundary of S.

Question 29

True or False: For every set S in respect to R, a point z must be either in int S, ext S, or ∂S.

Answer 29

True

The sets int S, ext S, and ∂S are pairwise disjoint.

Question 30

What is the relationship between int S, ext S, and ∂S?

Answer 30

The sets are pairwise disjoint: int S ∩ ext S = ∅, ext S ∩ ∂S = ∅, and int S ∩ ∂S = ∅.

The union of int S and ext S is R.

Question 31

Fill in the blank: The set of natural numbers N has ______ interior points.

Answer 31

∅

All points of N are boundary points.

Question 32

What is the interior of the set of integers Z?

Answer 32

int Z = ∅

All points of Z are boundary points.

Question 33

In the set of rational numbers Q, what can be said about its interior points?

Answer 33

int Q = ∅

Every neighbourhood of a rational number contains irrational numbers.

Question 34

What is the interior of the open interval (1, 2)?

Answer 34

int (1, 2) = (1, 2)

Every point in the interval is an interior point.

Question 35

What can be said about the boundary points of the closed interval [1, 2)?

Answer 35

∂[1, 2) = {1, 2}

Points 1 and 2 are boundary points.

Question 36

True or False: The empty set has interior points.

Answer 36

False

int ∅ = ∅.

Question 37

What is the exterior of the set of real numbers R?

Answer 37

ext R = ∅

Every point in R is an interior point.

Question 38

Fill in the blank: For a finite set S = {a1, a2, …, an}, the interior points are ______.

Answer 38

∅

All points in S are boundary points.

Question 39

What is the boundary of the set of real numbers R?

Answer 39

∂R = ∅

No point of R is a boundary point.