A description/proof of that image of continuous map from compact topological space to \(\mathbb{R}\) has minimum and maximum
Topics
About: topological space
About: map
The table of contents of this article
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.
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.
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.
Main Body
1: Description
For any compact topological space, T, \(\mathbb{R}\) with the Euclidean topology, and any continuous map, \(f: T \rightarrow \mathbb{R}\), \(f (T)\) has the minimum and the maximum.
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 \(f (T)\), \(inf f (T), sup f (T) \lt \infty\). \(inf f (T)\) 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 \(inf f (T)\) is an accumulation point, it is contained in \(f (T)\) any way, so, is the minimum. Likewise, \(sup f (T)\) is the maximum.
