# Why there are no constant functions on $\mathbb{R}$ with compact support?

For some the question might seem trivial but the concept is new to me and I have been wondering why there are no constant functions on $$\mathbb{R}$$ with compact support?

Following wiki:

Def.1) Functions with compact support on a topological space $$X$$ are those whose support is a compact subset of $$X$$.

Def.2) $$\operatorname{supp}f=:\overline{\{x\in X\;,f(x)\neq 0\}}$$

What if we take such function $$f:\mathbb{R}\to\mathbb{R}$$ defined as $$f(x)=0\;,\forall x\in\mathbb{R}$$ then $$\operatorname{supp}f=\overline{\{x\in\mathbb{R}\;,f(x)\neq 0\}}=\overline{\emptyset}=\emptyset$$ but $$\emptyset$$ is a compact subset of $$\mathbb{R}$$ I think....

Could someone enlighten me? Thank you.

• Where were you told that no constant functions on $\mathbb{R}$ have compact support? – Hayden May 31 '14 at 18:28
• well, anyway that's the only one there is, $f \equiv 0$ – mm-aops May 31 '14 at 18:30
• @Hayden: here for example www-personal.umich.edu/~wangzuoq/437W13/Notes/Lec%2030.pdf (12th line) – user124471 May 31 '14 at 18:32
• Well, I'd beg to differ with the claim, for precisely the example you gave. But as mm-aops pointed out, it's the only example. – Hayden May 31 '14 at 18:35
• the statement is very clear : ''there are NO functions'', (such functions don't exists). The real question here is if my counter-example is really correct (i.e that I followed all definitions correctly and so on). – user124471 May 31 '14 at 18:37

## 1 Answer

The author in the article you link to (www-personal.umich.edu/~wangzuoq/437W13/Notes/Lec%2030.pdf) is just being sloppy. He really should have said there are no non-zero constant functions with compact support on $\mathbb R$ (and thus $H_c^0(\mathbb R)$.

So yes, your example is correct. The empty set is (tautologically) compact.

• Hi Fredrik. Thank you. Unfortunately it is not the first time I have seen it....Exactly the same statement I found in Bott, Tu, ''Differential forms in algebraic topology''. Cheers – user124471 May 31 '14 at 18:44
• @user124471 Yep, I've seen it as well. In fact, I was just reading in the book by Bott & Tu a week ago, and had to think about the exact same question as you asked. – Fredrik Meyer May 31 '14 at 18:47
• @user124471 I was wondering the same thing! Now I know. Can you include Bott Tu in your question? – user636532 May 29 '19 at 11:27