any topological space with the fixed point property is connected
Theorem Any topological space^{} with the fixed-point property (http://planetmath.org/FixedPointProperty) is connected.
Proof. We will prove the contrapositive. Suppose $X$ is a topological space which is not connected. So there are non-empty disjoint open sets $A,B\subseteq X$ such that $X=A\cup B$. Then there are elements $a\in A$ and $b\in B$, and we can define a function $f:X\to X$ by
$$f(x)=\{\begin{array}{cc}a,\hfill & \text{when}x\in B,\hfill \\ b,\hfill & \text{when}x\in A.\hfill \end{array}$$ |
Since $A\cap B=\mathrm{\varnothing}$ and $A\cup B=X$, the function $f$ is well-defined. Also, $a\notin B$ and $b\notin A$, so $f$ has no fixed point. Furthermore, if $V$ is an open set in $X$, a short calculation shows that ${f}^{-1}(V)$ is $\mathrm{\varnothing},A,B$ or $X$, all of which are open sets. So $f$ is continuous, and therefore $X$ does not have the fixed-point property. $\mathrm{\square}$
References
- 1 G.J. Jameson, Topology^{} and Normed Spaces^{}, Chapman and Hall, 1974.
- 2 L.E. Ward, Topology, An Outline for a First Course, Marcel Dekker, Inc., 1972.
Title | any topological space with the fixed point property is connected |
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Canonical name | AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected |
Date of creation | 2013-03-22 13:56:35 |
Last modified on | 2013-03-22 13:56:35 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 12 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 47H10 |
Classification | msc 54H25 |
Classification | msc 55M20 |