The question reads: Is it true that every compact subset of [tex]\mathbb{R}[/tex] is the support of a continuous function? If not, can you describe the class of all compact sets in [tex]\mathbb{R}[/tex] which are supports of continuous functions? Is your description valid in other topological spaces?(adsbygoogle = window.adsbygoogle || []).push({});

The answer to the first question is no. A singleton is compact but is not the support of any continuous function; the same is true of the Cantor set (for it contains no segment). I am struck on the second question.

A continuous function is (in our text) defined as a function [tex]f:X\rightarrow Y[/tex] for topological spaces X and Y such that [tex]f^{-1}\left( V\right)[/tex] is an open set in X for every open set V in Y.

The support of a function is the closure of the set set of all values at which it is not zero, that is [tex]\overline{\left\{ x:f(x) \mbox{ not }= 0\right\} }[/tex]

So I need to describe

[tex]\left\{ K\subset \mathbb{R}: \exists \mbox{ a continuous function } f \mbox{ such that support}(f)=K\right\} \cap \left\{ K\subset \mathbb{R}:K \mbox{ is compact} \right\}[/tex]

Based on the Cantor set example given as a counter-example to the first question, I'm guessing that connectedness may be involved, but I really don't know. How can I answer this question so that it holds for a general topological space?

Please help.

-Ben

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Describe the set of all compact sets which are supports of continuous functions

**Physics Forums | Science Articles, Homework Help, Discussion**