# Topological Terms

Hey there,Here’s discussing some basic topological terms such as the most useful definitions like T_{0} 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:

#### T_{0} Space:

We can define a topological space T_{0} 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.

#### T_{1} Space:

In topology, the T_{1} 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.

#### T_{2} Space (Hausdorff Space):

In topology,T_{2} 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 T_{2} Space the Hausdorff space.

### Important notes about topological spaces:

- T
_{2}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=R^{n} denotes the set of all ordered n-tuples of real numbers for fixed n=N.

if x,y ∈ R^{n} , then

x=<x_{1},x_{2},—x_{n}> , y=<y_{1},y_{2},—y_{n}>

Define a mapping

d:R^{n}xR^{n}→R

d[E^{n}(x,y)] = dE^{n}(<x_{1},x_{2},—x_{n}><y_{1},y_{2},—y_{n}>)

= √∑_{i=1}^{∞}(x_{i}-y_{i})^{2}

The set (R^{n},dE^{n}) is the real Euclidean space.

#### Hilbert space:

Let F denotes the set of all infinite sequences i.e.if x F then, x=<x_{1},x_{2},—x_{n}> of real numbers for which ∑_{i=1}^{∞} x^{2} converges.

Let x,y ∈ H then,

x=<x_{1},x_{2},—x_{n}>

y=<y_{1},y_{2},—y_{n}>

Then, d_{11}(x,y)=d_{h}(<x_{1},x_{2},—x_{n}><y_{1},y_{2},—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.