- #1

benorin

Homework Helper

- 1,344

- 136

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?

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

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

Last edited: