Editing General topology (section)

{{Short description|Branch of topology}}
[[Image:Topologist's sine curve.svg|thumb|420px|The [[Topologist's sine curve]], a useful example in point-set topology. It is connected but not path-connected.]]

In [[mathematics]], '''general topology''' (or '''point set topology''') is the branch of [[topology]] that deals with the basic [[Set theory|set-theoretic]] definitions and constructions used in topology. It is the foundation of most other branches of topology, including [[differential topology]], [[geometric topology]], and [[algebraic topology]].

The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': 
* [[Continuous function]]s, intuitively, take nearby points to nearby points. 
* [[Compact set]]s are those that can be covered by finitely many sets of arbitrarily small size.
* [[Connected set]]s are sets that cannot be divided into two pieces that are far apart. 
The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of [[open set]]s. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''topology''. A set with a topology is called a ''[[topological space]]''.

''[[Metric space]]s'' are an important class of topological spaces where a real, non-negative distance, also called a ''[[metric (mathematics)|metric]]'', can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.