Topology Basics Flashcards
What are intervals in the context of the ordered field structure of R?
Intervals are special subsets of R.
What is the definition of an open interval (a, b)?
The open interval is the set (a, b) := {x ∈ R : a < x < b}.
What is the definition of a closed interval [a, b]?
The closed interval is the set [a, b] := {x ∈ R : a ≤ x ≤ b}.
What is the definition of a half-open (or half-closed) interval?
The half-open intervals are defined as (a, b) := {x ∈ R : a < x < b} and [a, b) := {x ∈ R : a ≤ x < b}.
What are the endpoints of an interval?
The points a and b are called the endpoints of the interval.
What is the length of an interval?
Each of the intervals is bounded and has length defined by b - a.
What happens if a = b in terms of intervals?
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}.
How are unbounded intervals defined?
Unbounded intervals can be defined by replacing one or both endpoints with the symbols +∞ and -∞.
What are the definitions of infinite open intervals?
The infinite open intervals are defined as (a, ∞) := {x ∈ R : x > a} and (-∞, b) := {x ∈ R : x < b}.
What are the definitions of infinite closed intervals?
The infinite closed intervals are defined as [a, ∞) := {x ∈ R : x ≥ a} and (-∞, b] := {x ∈ R : x ≤ b}.
What is the definition of the infinite interval (-∞, ∞)?
The infinite interval is the entire set R, i.e., (-∞, ∞) := R.
What is the definition of an 𝜖-neighbourhood of a point c?
The set of all points in the open interval (c - 𝜖, c + 𝜖)
Denoted as N𝜖(c) for every 𝜖 > 0.
What does it mean for a set S to be a neighbourhood of a point c?
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.
Is the closed interval [1, 2] a neighbourhood of the point 1?
No
Although it contains the point 1, it is not an open interval.
What is the result of the union of two neighbourhoods of a point?
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.
True or False: The intersection of two neighbourhoods of a point is also a neighbourhood of the point.
True
If S1 and S2 are neighbourhoods of c, then S1 ∩ S2 is a neighbourhood of c.
What happens when you take the union of a finite number of neighbourhoods of a point?
It is also a neighbourhood of the point
This follows from the conclusion about the union of two neighbourhoods.
What happens when you take the intersection of a finite number of neighbourhoods of a point?
It is also a neighbourhood of the point
This is based on the conclusion regarding the intersection of two neighbourhoods.
What is the result of the union of an infinite number of neighbourhoods of a point?
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.
Can the intersection of an infinite number of neighbourhoods of a point be a neighbourhood of that point?
No, it may not be
For example, the intersection of intervals that shrink down to a point may not contain any open intervals.
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 _______.
(c - n, c + n) ⊆ S
What is the example of a 0.001-neighbourhood of the point 5?
(4.999, 5.001)
This is derived from the definition of 𝜖-neighbourhood.
What is an interior point of a set S?
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).
What is the notation for the set of all interior points of S?
int S, Sᵢ, or S°
The interior of S is always a subset of S.
What defines an exterior point of a set S?
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 = ∅.
What is the notation for the set of all exterior points of S?
ext S or Sᵉ
The exterior of S is always a subset of the complement of S.
How is a boundary point of S defined?
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.
What is the notation for the set of all boundary points of S?
∂S
This notation represents the boundary of S.
True or False: For every set S in respect to R, a point z must be either in int S, ext S, or ∂S.
True
The sets int S, ext S, and ∂S are pairwise disjoint.
What is the relationship between int S, ext S, and ∂S?
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.
Fill in the blank: The set of natural numbers N has ______ interior points.
∅
All points of N are boundary points.
What is the interior of the set of integers Z?
int Z = ∅
All points of Z are boundary points.
In the set of rational numbers Q, what can be said about its interior points?
int Q = ∅
Every neighbourhood of a rational number contains irrational numbers.
What is the interior of the open interval (1, 2)?
int (1, 2) = (1, 2)
Every point in the interval is an interior point.
What can be said about the boundary points of the closed interval [1, 2)?
∂[1, 2) = {1, 2}
Points 1 and 2 are boundary points.
True or False: The empty set has interior points.
False
int ∅ = ∅.
What is the exterior of the set of real numbers R?
ext R = ∅
Every point in R is an interior point.
Fill in the blank: For a finite set S = {a1, a2, …, an}, the interior points are ______.
∅
All points in S are boundary points.
What is the boundary of the set of real numbers R?
∂R = ∅
No point of R is a boundary point.
