# Point-set topology

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**Point-set topology** is a fundamental branch of topology, sometimes referred to as **general topology**, which deals with the concepts of **topological spaces** and the mathematical structures defined on such spaces.

## Topology and open sets

Given a set , a family of subsets of is said to be a **topology** of if the following three conditions hold:

- (The empty set and are both elements of )
- (Any union of elements of is an element )
- (Any finite intersection of elements of is an element of )

The members of a topology are called **open sets** of the topology.

## Topological space

A **topological space** is a set , known as the underlying set, together with a topology T of .

## Basis for a topology

A **basis** for a topology on is a collection of subsets of , known as basis elements, such that the following two properties hold:

- For every there is at least one basis element that contains .
- If is an element of the intersection of two basis elements , then there exists a basis element such that .

Given a basis for a topology, one can define the **topology generated by the basis** as the collection of all sets such that for each there is a basis element such that and .

## Closed sets

A set is defined to be **closed** if its complement in is an open set in the given topology.

## Neighborhoods

A set is said to be a **neighborhood** of a point if it is an open set which contains the point . In some cases the term neighborhood is used to describe a set which contains an open set containing .

## Interior and closure

The **interior** of a subset of is defined to be the union of all open sets contained in .

The **closure** of a subset of is defined as the intersection of all closed sets containing .

## Limit points

A point of is said to be a **limit point** of a subset A of if every neighborhood of intersects A in at least one point other than .

## Continuous functions

A function is said to be **continuous** if for each subset of , the set is an open set of .

## Homeomorphisms

A bijective function is said to be a **homeomorphism** if both and its inverse, , are continuous.

If there exists a homeomorphism between to topological spaces X and Y, then the spaces are said to be **homeomorphic**.

Any property that is invariant under homeomorphisms is known as a **topological property**.

A homeomorphism is also dubbed a **topological equivalence** among mathematicians.