# Topology of Open Sets

A **topology** T on an open set X is a collection of subsets of X deemed "open" that satisfies the following properties:

- The empty set Ø and the set X itself are open by definition.
- The union and intersection of any finite number of open sets are also open.

In other words, the collection T must only include subsets of X considered "open" and is closed under the operations of union and intersection.

Where a collection of sets is a set whose elements are sets or subsets.

In these instances, the set X and the topology T constitute a **topological space**, often denoted as the pair (X,T).

**Note**. For simplicity, it's often said that X is a topological space. However, it's important to remember that a topological space consists of two elements: the topology T (collection of subsets) and the set X.

**Why is the empty set always considered an open set?**

The empty set is considered open by definition in every topological space.

This convention is a fundamental part of topology definitions and ensures the properties and axioms of topology are consistent and complete.

## A Practical Example

Consider a set X with three elements A, B, C

$$ X = \{ A,B,C \} $$

A topology T could be the collection of subsets { }, {A,B,C}, {B}, {B,C}

$$ T = \{ \{ \},\{A,B,C \}, \{ B \}, \{ B,C \} \} $$

Where { } represents the empty set Ø, and {A,B,C} is the set itself, or the improper subsets of set X.

By definition, the empty set and the complete set X are open sets.

A topology must be composed of open sets and, by definition, the union and intersection of open sets yield another open set.

In this case, the union of subsets considered in T still belongs to the collection T.

Thus, the collection T is closed to union operations.

$$ \{ B \} \cup \{ B, C \} \subseteq \{ B, C \} \subseteq T$$

$$ \{ B \} \cup \{ A, B, C \} \subseteq \{ A, B, C \} \subseteq T$$

$$ \{ B \} \cup \{ \} \subseteq \{ B \} \subseteq T $$

$$ \{ B \} \cup \{ B \} \subseteq \{ B \} \subseteq T$$

Furthermore, the intersection of subsets in T remains part of the collection T.

Therefore, the collection T is also closed to intersection operations.

$$ \{ B \} \cap \{ B, C \} \subseteq \{ B \} \subseteq T$$

$$ \{ B \} \cap \{ A, B, C \} \subseteq \{ B \} \subseteq T$$

$$ \{ B \} \cap \{ \} \subseteq \{ \} \subseteq T $$

$$ \{ B \} \cap \{ B \} \subseteq \{ B \} \subseteq T$$

Therefore, the collection T is a topology of set X because it meets all necessary conditions.

**Example 2**

Consider a slightly different collection from the previous example.

The set X remains the same.

$$ X = \{ A,B,C \} $$

In this case, however, the collection T also includes the subset {A} in addition to the subsets { }, {A,B,C}, {B}, {B,C}.

$$ T = \{ \{ \},\{A,B,C \}, \{ A \}, \{ B \}, \{ B,C \} \} $$

This new collection T of subsets __is not a topology__ of set X because it does not satisfy all conditions.

For example, the union of subsets {A} and {B} creates a set {A,B} that is not included in the collection T itself.

$$\require{cancel} \{ A \} \cup \{ B \} = \{ A, B \} \cancel{\in} T $$

In other words, the two sets {A} and {B} are considered "open" sets because they are found in the collection T.

However, their union does not create another open set {A,B} because the set {A,B} is not found in the collection.

$$ T = \{ \{ \},\{A,B,C \}, \{ A \}, \{ B \}, \{ B,C \} \} $$

This violates one of the necessary conditions for the topology of an open set.

Therefore, **the collection T cannot be considered a topology of set X**.

And so on