# 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 X, a family of subsets T of X is said to be a **topology** of X if the following three conditions hold:

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

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

## Topological space

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

## Basis for a topology

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

- For every
*x*in X there is at least one basis element B that contains*x*. - If
*x*is an element of the intersection of two basis elements A and B, then there exists a basis element C such that .

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

## Closed sets

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

## Neighborhoods

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

## Interior and closure

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

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

## Limit points

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

## 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.