Topology :(

A collection of subsets of
X
X, called open sets, satisfying the topology axioms.
Which two sets must always be open in a topology on
X

The empty set
∅
∅ and the entire set
X
X.

The intersection of finitely many open sets is open.

The union of any collection of open sets is open.

A set
X
X together with a topology
T
T.
What is the trivial topology on a nonempty set
X

The topology
T
=
{
∅
,
X
}
T={∅,X}.

The trivial topology.
What is the discrete topology on a set
X

The topology consisting of all possible subsets of
X
X.
Which topology is the largest possible topology on a set
X

The discrete topology.
What is the finite complement topology
R
f
c
R
fc
​

The topology on
R
R whose open sets are
∅
∅ and all subsets whose complements are finite.
Which subsets are open in the finite complement topology on
R

The empty set and every subset of
R
R with a finite complement.
When is a topology
T
2
T
2
​
finer than a topology
T
1
T
1
​

When
T
1
⊂
T
2
T
1
​
⊂T
2
​
.
When is a topology
T
1
T
1
​
coarser than a topology
T
2
T
2
​

When
T
1
⊂
T
2
T
1
​
⊂T
2
​
.
When is
T
2
T
2
​
strictly finer than
T
1
T
1
​

When
T
1
⊂
T
2
T
1
​
⊂T
2
​
and
T
1
≠
T
2
T
1
​


=T
2
​
.
What is a neighborhood of a point
x

An open set
U
U containing
x
x.
What is a basis
B
B for a topology on
X

A collection of subsets of
X
X satisfying the basis axioms.

Every point of
X
X belongs to at least one basis element.

If
x
∈
B
1
∩
B
2
x∈B
1
​
∩B
2
​
for basis elements
B
1
,
B
2
B
1
​
,B
2
​
, there exists
B
3
∈
B
B
3
​
∈B such that
x
∈
B
3
⊂
B
1
∩
B
2
x∈B
3
​
⊂B
1
​
∩B
2
​
.

The subsets belonging to a basis
B
B.

Its open sets are the empty set and arbitrary unions of basis elements.
What is the standard topology on
R

The topology generated by all open intervals
(
a
,
b
)
(a,b).
What basis generates the standard topology on
R

The collection of all open intervals
(
a
,
b
)
(a,b) in
R
R.
What is the lower limit topology
R
l
R
l
​

The topology on
R
R generated by the basis of half-open intervals
[
a
,
b
)
[a,b).

Half-open intervals of the form
[
a
,
b
)
[a,b).

The set of integers
Z
Z.
What is the digital line basis element
B
(
n
)
B(n) when
n

B
(
n
)
=
{
n
}
B(n)={n}.
What is the digital line basis element
B
(
n
)
B(n) when
n

B
(
n
)
=
{
n
−
1
,
n
,
n
+
1
}
B(n)={n−1,n,n+1}.
What generates the standard topology on
R
2
R
2
and
R
n
R
n

The collection of all open balls defined using Euclidean distance.
What is an open ball
B
(
x
,
ϵ
)

The set of points whose Euclidean distance from
x
x is less than
ϵ
ϵ.
When is a subset
A
A of a topological space
X

When its complement
X
−
A
X−A is open.
How is the complement of a subset
A
A of
X

X
−
A
X−A or
X
∖
A
X∖A.
What is a closed ball in
R
2
R
2

B
ˉ
(
x
,
ϵ
)
=
{
y
∈
R
2
∣
d
(
x
,
y
)
≤
ϵ
}
B
ˉ
(x,ϵ)={y∈R
2
∣d(x,y)≤ϵ}.

d
(
x
,
y
)
≤
ϵ
d(x,y)≤ϵ.
What is a closed rectangle in
R
2
R
2

A product of closed bounded intervals,
[
a
,
b
]
×
[
c
,
d
]
[a,b]×[c,d].

A topological space in which every pair of distinct points has disjoint neighborhoods containing the respective points.

There must exist disjoint neighborhoods containing the two points.
What is the interior of a subset
A

The union of all open sets contained in
A
A.
What symbols denote the interior of
A

A
˚
A
˚
or
Int
⁡
(
A
)
Int(A).
What is the closure of a subset
A

The intersection of all closed sets containing
A
A.
What symbols denote the closure of
A

A
ˉ
A
ˉ
or
Cl
⁡
(
A
)
Cl(A).
What is a dense subset
A
A of a topological space
X

A subset whose closure equals
X
X, so
Cl
⁡
(
A
)
=
X
Cl(A)=X.
What is the defining condition for a point
x
x to be a limit point of
A

Every neighborhood of
x
x intersects
A
A in a point other than
x
x.
Can a limit point of
A
A belong to
A

Yes. A limit point may or may not belong to
A
A.
When does a sequence
(
x
n
)
(x
n
​
) converge to
x

When every neighborhood
U
U of
x
x contains all terms
x
n
x
n
​
for sufficiently large
n
n.
What does convergence of a sequence require for every neighborhood
U
U of
x

There must exist an integer
N
N such that
x
n
∈
U
x
n
​
∈U for all
n
≥
N
n≥N.

lim
⁡
n
→
∞
x
n
=
x
lim
n→∞
​
x
n
​
=x.
What is the boundary of a subset
A

The closure of
A
A minus its interior.
What symbol denotes the boundary of
A

∂
A
∂A.
What is the formula for the boundary of a set
A

∂
A
=
Cl
⁡
(
A
)
−
Int
⁡
(
A
)
∂A=Cl(A)−Int(A).

The boundary is the part of the closure that is not in the interior.