generalized intermediate value theorem
Theorem.
Let be a continuous function with a connected space and a totally ordered set in the order topology. If and lies between and , then there exists such that .
Proof.
The sets and are disjoint open subsets of in the subspace topology, and they are both non-empty, as is contained in one and is contained in the other. If , then constitutes a of the space , contradicting the hypothesis that is the continuous image of the connected space . Thus there must exist such that . ∎
This version of the intermediate value theorem reduces to the familiar one of http://planetmath.org/node/7599real analysis when is taken to be a closed interval in and is taken to be .
References
- 1 J. Munkres, Topology, 2nd ed. Prentice Hall, 1975.
Title | generalized intermediate value theorem |
---|---|
Canonical name | GeneralizedIntermediateValueTheorem |
Date of creation | 2013-03-22 17:17:44 |
Last modified on | 2013-03-22 17:17:44 |
Owner | azdbacks4234 (14155) |
Last modified by | azdbacks4234 (14155) |
Numerical id | 8 |
Author | azdbacks4234 (14155) |
Entry type | Theorem |
Classification | msc 26A06 |
Related topic | OrderTopology |
Related topic | TotalOrder |
Related topic | Continuous |
Related topic | ConnectedSpace |
Related topic | ConnectednessIsPreservedUnderAContinuousMap |