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.

  • $\begingroup$ Where were you told that no constant functions on $\mathbb{R}$ have compact support? $\endgroup$ – Hayden May 31 '14 at 18:28
  • $\begingroup$ well, anyway that's the only one there is, $f \equiv 0$ $\endgroup$ – mm-aops May 31 '14 at 18:30
  • $\begingroup$ @Hayden: here for example www-personal.umich.edu/~wangzuoq/437W13/Notes/Lec%2030.pdf (12th line) $\endgroup$ – user124471 May 31 '14 at 18:32
  • $\begingroup$ 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. $\endgroup$ – Hayden May 31 '14 at 18:35
  • $\begingroup$ 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). $\endgroup$ – user124471 May 31 '14 at 18:37

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.

  • 2
    $\begingroup$ 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 $\endgroup$ – user124471 May 31 '14 at 18:44
  • 1
    $\begingroup$ @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. $\endgroup$ – Fredrik Meyer May 31 '14 at 18:47
  • $\begingroup$ @user124471 I was wondering the same thing! Now I know. Can you include Bott Tu in your question? $\endgroup$ – Selene Auckland May 29 at 11:27

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