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Main article: General topology General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.
Another name for general topology is point-set topology. The basic object of study is topological spaces , which are sets equipped with a topology , that is, a family of subsets , called open sets, which is closed under finite intersections and finite or infinite unions. The fundamental concepts of topology, such as continuity , compactness , and connectedness , can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points.
Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart.
The words nearby, arbitrarily small, and far apart can all be made precise by using open sets.
Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets, and this changes which functions are continuous, and which subsets are compact or connected.
Metric spaces are an important class of topological spaces where distances between any two points are defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of the points whose distance to x is less than d.
Many common spaces are topological space whose topology can be defined by a metric. This is the case of the real line , the complex plane , real and complex vector spaces and Euclidean spaces.
Having a metric simplifies many proofs. Main article: Algebraic topology Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The most important of these invariants are homotopy groups , homology, and cohomology. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.
Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main article: Differential topology Differential topology is the field dealing with differentiable functions on differentiable manifolds.
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