**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 $ \tau $ of $ X $ is said to be a **topology** of $ X $ if the following three conditions hold:

- $ X,\varnothing\in\tau $ (The empty set and $ X $ are both elements of $ \tau $)
- $ \{A_i\}_{i\in I}\in\tau\rArr\bigcup_{i\in I}A_i\in\tau $ (Any union of elements of $ \tau $ is an element $ \tau $)
- $ A,B\in\tau\rArr A\cap B\in\tau $ (Any finite intersection of elements of $ \tau $ is an element of $ \tau $)

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,B $ , then there exists a basis element $ C $ such that $ C\subset A\cap B $ .

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 $ x\in A $ there is a basis element $ B $ such that $ x\in B $ and $ B\subset A $.

## 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 $ f:X\to Y $ is said to be **continuous** if for each subset $ A $ of $ Y $ , the set $ f^{-1}(A) $ is an open set of $ X $ .

## Homeomorphisms

A bijective function $ f:X\to Y $ is said to be a **homeomorphism** if both $ f $ and its inverse, $ f^{-1}:Y\to X $ , are continuous.

If there exists a homeomorphism between two 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.