البوابة
[edit] Mathematical definition
[edit] Some theorems in general topology
Main article: Topological space
Let X be any set and let T be a family of subsets of X. Then T is a topology on X if
Let X be any set and let T be a family of subsets of X. Then T is a topology on X if
- Both the empty set and X are elements of T.
- Any union of arbitrarily many elements of T is an element of T.
- Any intersection of finitely many elements of T is an element of T.
If T is a topology on X, then X together with T is called a topological space.
All sets in T are called open; note that in general not all subsets of X need be in T. A subset of X is said to be closed if its complement is in T (i.e., it is open). A subset of X may be open, closed, both, or neither.
A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both space with the Standard Topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered to be topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut.
All sets in T are called open; note that in general not all subsets of X need be in T. A subset of X is said to be closed if its complement is in T (i.e., it is open). A subset of X may be open, closed, both, or neither.
A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both space with the Standard Topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered to be topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut.
[edit] Some theorems in general topology
- Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closed and bounded. (See Heine-Borel theorem).
- Every continuous image of a compact space is compact.
- Tychonoff's theorem: The (arbitrary) product of compact spaces is compact.
- A compact subspace of a Hausdorff space is closed.
- Every sequence of points in a compact metric space has a convergent subsequence.
- Every interval in R is connected.
- The continuous image of a connected space is connected.
- A metric space is Hausdorff, also normal and paracompact.
- The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
- The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
- The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
- On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
- Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.
General topology also has some surprising connections to other areas of mathematics. For example:
- in number theory, Furstenberg's proof of the infinitude of primes.
