# Point-set topology

1,010pages on
this wiki

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:

1. $X,\emptyset \in \tau$ (The empty set and $X$ are both elements of $\tau$)
2. $\{A_i\}_{i \in I}\in \tau \Rightarrow \displaystyle\bigcup_{i \in I}A_i \in \tau$ (Any union of elements of $\tau$ is an element $\tau$)
3. $A,B \in \tau \Rightarrow A \cap B \in \tau$ (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:

1. For every x in X there is at least one basis element B that contains x.
2. If x is an element of the intersection of two basis elements A and 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 $f: X \rightarrow Y$ 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 $f: X \rightarrow Y$ is said to be a homeomorphism if both $f$ and its inverse, $f^{-1}:Y \rightarrow X$, 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.