In which of the following cases is there no continuous function $f$ from the set $S$ onto the set $T$?

Asked Jul 16 '13 at 07:42

Active Jul 16 '13 at 08:12

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In which of the following cases is there no continuous function $f$ from the set $S$ onto the set $T$?

$S=[0,1],T=\Bbb R$
$S=(0,1),T=\Bbb R$
$S=(0,1),T=(0,1]$
$S=\Bbb R,T=(0,1)$ how we solve it.plz explain

edited Jul 16 '13 at 08:12

asked Jul 16 '13 at 07:42

sam

125

You really need to figure out how to switch off caps lock. – Tobias Kildetoft Jul 16 '13 at 07:43
People generally interpret all capitals as SHOUTING; it’s not a good idea. – Brian M. Scott Jul 16 '13 at 07:45
1

Do you have any insight, idea, work done...in these? Say, compact sets are mapped to compact sets by cont. functions? – DonAntonio Jul 16 '13 at 07:48

1 Answers1

2

HINTS:

Continuous functions take compact sets to compact sets.
It’s not hard to find a homeomorphism; you could start by thinking about the function $f(x)=\tan x$ and then modifying it a bit.
Fold $S$ in the middle and then stretch it by a factor of $2$.
If you’ve done (2), this one is easy.

answered Jul 16 '13 at 07:48

Brian M. Scott

616,228

1

I especially like the nice geometric view of number 3. – Tobias Kildetoft Jul 16 '13 at 07:54