316: Image of Continuous Map from Compact Topological Space to \mathbb{R} Has Minimum and Maximum

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A description/proof of that image of continuous map from compact topological space to $\mathbb{R}$ has minimum and maximum

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Topics

About: topological space
About: map

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of compact topological space.
• The reader knows a definition of Euclidean topology.
• The reader knows a definition of continuous map.
• The reader admits the proposition that the continuous image of any compact topological space is compact.
• The reader admits the Heine-Borel theorem: any subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded.
• The reader admits the proposition that any subset is closed if and only if it equals its closure.
• The reader admits the proposition that the closure of any subset is the union of the subset and the set of all the accumulation points of the subset.

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Target Context

• The reader will have a description and a proof of the proposition that the image of any continuous map from any compact topological space to $\mathbb{R}$ with the Euclidean topology has the minimum and the maximum.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any compact topological space, T, $\mathbb{R}$ with the Euclidean topology, and any continuous map, $f:T\to \mathbb{R}$, $f(T)$ has the minimum and the maximum.

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2: Proof

By the proposition that the continuous image of any compact topological space is compact, $f(T)$ is compact on $\mathbb{R}$. By the Heine-Borel theorem: any subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, $f(T)$ is closed and bounded. As $f(T)$ is bounded, there are the infimum and the supremum of $f(T)$, $inff(T),supf(T)<\mathrm{\infty }$. $inff(T)$ is the minimum or an accumulation point of $f(T)$, but as $f(T)$ is closed, by the proposition that any subset is closed if and only if it equals its closure, $f(T)=\overline{f(T)}$ where $\overline{f(T)}$ is the closure of $f(T)$, but by the the proposition that the closure of any subset is the union of the subset and the set of all the accumulation points of the subset, $\overline{f(T)}=f(T)\cup ac(f(T))$ where $ac(f(T))$ is the set of all the accumulation points of $f(T)$, so, if $inff(T)$ is an accumulation point, it is contained in $f(T)$ any way, so, is the minimum. Likewise, $supf(T)$ is the maximum.

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References

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