Vipul: Created page with '==Statement== Suppose $X$ is a topological space that satisfies the conclusion of the intermediate value theorem: For any continuous function $f:X \to \R…'$

2009-12-24T01:04:05Z

Created page with '==Statement== Suppose <math>X</math> is a topological space that satisfies the conclusion of the intermediate value theorem: For any continuous function <math>f:X \to \R…'

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==Statement==

Suppose <math>X</math> is a [[topological space]] that satisfies the conclusion of the [[intermediate value theorem]]: For any continuous function <math>f:X \to \R</math>, and two elements <math>x_1,x_2 \in X</math> such that <math>f(x_1) < f(x_2)</math>, <math>f(X)</math> must contain <math>[f(x_1),f(x_2)]</math>.

Then, <math>X</math> is a [[connected space]].

==Related facts==

===Converse===

* [[Intermediate value theorem]]
==Proof==

'''Given''': A topological space <math>X</math> such that for any continuous function <math>f:X \to \R</math>, and two elements <math>x_1,x_2 \in X</math> such that <math>f(x_1) < f(x_2)</math>, <math>f(X)</math> must contain <math>[f(x_1),f(x_2)]</math>.

'''To prove''': <math>X</math> is connected.

'''Proof''': Suppose not, i.e., suppose <math>X</math> is not connected. Then, <math>X</math> is a union of two nonempty disjoint open subsets <math>U</math> and <math>V</math>. Consider the function <math>f:X \to \R</math> defined by <math>f(x) = 0 \ \forall \ x \in U</math> and <math>f(x) = 1 \ \forall \ x \in V</math>. This is a continuous function (in fact, all its fibers are open).

Pick <math>x_1 \in U</math> and <math>x_2 \in V</math>. By our construction, <math>f(x_1) < f(x_2)</math>, so by the given data, <math>f(X)</math> should contain the interval <math>[f(x_1),f(x_2)] = [0,1]</math>. But this contradicts the fact that the image of <math>f</math> is the two-element set <math>\{ 0, 1 \}</math>.

Thus, our original assumption that <math>X</math> is not connected cannot hold. Hence, <math>X</math> must be connected.

Converse of intermediate value theorem - Revision history

Vipul: Created page with '==Statement== Suppose X is a topological space that satisfies the conclusion of the intermediate value theorem: For any continuous function f:X \to \R…'

Vipul: Created page with '==Statement== Suppose $X$ is a topological space that satisfies the conclusion of the intermediate value theorem: For any continuous function $f:X \to \R…'$