Topological Terms

Hey there,Here’s discussing some basic topological terms such as the most useful definitions like T₀ spaces, open sets, ε-net… that are the backbone of the subject topology.

Topological Spaces:

A topological space (X,ℑ) is a set, X of points &a family of subsets of X which satisfy the following property:

• The union of any number of members of ℑ is a member of ℑ.
• The intersection of any finite number of members of ℑ is a member of ℑ .
• Φ & X∈(X,ℑ ); Where members of ℑ are open sets.

Indescrete Topology:

Topology containing only Φ & X; where Φ is null set.&X is the whole set.

Descrete Topology:

Topology consisting of other elements along withΦ & X; where Φ is null set.&X is the whole set.

• Indescrete topology is the smallest topology. i.e. Weakest topology.
• Descrete topology is the strongest topology. i.e. Coarcest topology.
• Union & intersection of any no. of topology is a topology.

 

Limit or Accumulation point:

x is a limit point of a subset E if & only if every open set containing x contains a point of E different from x i.e.if x∈G∈ℑ, then E∩G\{x}≠Φ.

Interior point:

Specifically, in topology,A point that is in the interior of the set A is the interior point.

Derived set in a topology:

In topology, the set of all limit points is the derived set.

Some tips for Derived sets:

• d(Φ) = Φ
• If A ⊆ B then d(A) ⊆ d(B)
• If x∈d(E), then x∈d(E\{x})
• d(A∪B)=d(A)∪d(B)

 Open set:

In topology,A set containing all its interior point, is the open set.

Closed Set:

In topology,A set containing all its limit points, is the closed set.

Topological Spaces:

T₀ Space:

We can define a topological space T₀ Space as:
If x &y are two distinct points of the set X, then there exist an open set which contain one of them but not the other.

T₁ Space:

In topology, the T₁ Space is defined as:                                                                    If x &y are two distinct points of the set X, then there exist two open sets one containing x but not y and other containing y but not x.

T₂ Space (Hausdorff Space):

In topology,T₂ Space is defined as:                                                                          If x &y are two distinct points of the set X, then there exist two open sets one containing x and other containing y.                                                                           Another name of T₂ Space the Hausdorff space.

Important notes about topological spaces:

• T₂ Space is always a T1 Space & T0 Space,i.e. it always satisfy axioms of T0 Space.
• T1 Space is always a T0 Space. i.e. it always satisfy axioms of T0 Space.
• But T0 Space can never be T1 Space & T2 Space.

First axiom space:

A topological space X is of first axiom space, if it satisfies the following axiom of countability:

For every point x∈X; There exist a countable family {B_n(x)} of open sets containing x such that whenever x ∈ open set G.
B_n(x) ∈ G for some n.

Second axiom space:

A topological space X is of second axiom space if and only if it satisfies the following axiom of countability:

There exist a countable base for topology τ.

ε-net:

Let(X,d) be a metric space & ε > 0. A finite subset E of X is said to be ε-net for X if and only if E is finite & for every x ∈ X there exist a point e E s.t. d(e,x)<ε.
To illustrate the above definition we make use of the diagram,

Here,square denotes space(X,d), The circle in figure denotes the subset of X, e is the point in E.

Lindelof Space:

Lindelof space is a topological space in which every open cover has a countable subcover.

Euclidean n-space:

Let X=Rⁿ denotes the set of all ordered n-tuples of real numbers for fixed n=N.
if x,y ∈ Rⁿ , then
x=<x₁,x₂,—x_n> , y=<y₁,y₂,—y_n>
Define a mapping
d:RⁿxRⁿ→R
d[Eⁿ(x,y)] = dEⁿ(<x₁,x₂,—x_n><y₁,y₂,—y_n>)
= √∑_i=1^∞(x_i-y_i)²
The set (Rⁿ,dEⁿ) is the real Euclidean space.

Hilbert space:

Let F denotes the set of all infinite sequences i.e.if x F then, x=<x₁,x₂,—x_n> of real numbers for which ∑_i=1^∞ x² converges.
Let x,y ∈ H then,
x=<x₁,x₂,—x_n>
y=<y₁,y₂,—y_n>
Then, d₁₁(x,y)=d_h(<x₁,x₂,—x_n><y₁,y₂,—y_n>)
= √∑_i=1^∞(x_i-y_i)²
Then the space (H,d_h) is metric space called Hilbert space.

Principle of Transfinite induction:

If X is the well ordered set & E is the subset of X with the property that Xx ⊆ E implies that x ∈ E, then E=X.

Axiom of choice:

Cartesion product of non empty family of non empty family of non empty of non empty sets is non empty.

Burali-forti paradox:

There is no set containing all the ordinal numbers.

Finite intersection property:

We define a finite intersection property as -every finite subfamily of the family has a non empty intersection.

Here, possibly all major concepts of topology are discussed.Hopefully, these would be useful yo the reader.