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 Edit

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:

1. The empty set and X are both elements of T
2. Any union of elements of T is an element T
3. 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 Edit

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

Basis for a topology Edit

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 Edit

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

Neighborhoods Edit

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 Edit

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 Edit

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 Edit

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 Edit

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.