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From: William Elliot on 26 Jul 2010 05:25
Let S be a space and p a point in S.

What topology can S have if a function f:S -> S
is continuous iff f(p) = p?

Is the topology unique?
From: G. A. Edgar on 26 Jul 2010 05:53
In article <20100726022448.E78483(a)agora.rdrop.com>, William Elliot
<marsh(a)rdrop.remove.com> wrote:

> Let S be a space and p a point in S.
>
> What topology can S have if a function f:S -> S
> is continuous iff f(p) = p?
>
> Is the topology unique?

Every constant function is continuous. So your condition means that S
has no point other than p.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
From: Butch Malahide on 26 Jul 2010 05:57
On Jul 26, 4:25 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> Let S be a space and p a point in S.
>
> What topology can S have if a function f:S -> S
>         is continuous iff f(p) = p?
>
> Is the topology unique?

Inasmuch as constant functions are continuous, p must be the only
point in the space. Yes, the topology on a one-point space is unique.
From: Bill on 26 Jul 2010 06:04
Butch Malahide wrote:
> On Jul 26, 4:25 am, William Elliot<ma...(a)rdrop.remove.com> wrote:
>> Let S be a space and p a point in S.
>>
>> What topology can S have if a function f:S -> S
>> is continuous iff f(p) = p?
>>
>> Is the topology unique?
>
> Inasmuch as constant functions are continuous, p must be the only
> point in the space. Yes, the topology on a one-point space is unique.

I think the OP means that f is the identity. The case |S|=1 appears to
be settled. What happens if |S|=2 (rhetorical)?

Bill
From: William Elliot on 26 Jul 2010 06:31
On Mon, 26 Jul 2010, G. A. Edgar wrote:
> <marsh(a)rdrop.remove.com> wrote:
>
>> Let S be a space and p a point in S.
>>
>> What topology can S have if a function f:S -> S
>> is continuous iff f(p) = p?
>>
>> Is the topology unique?
>
> Every constant function is continuous. So your condition means that S
> has no point other than p.
>
False. See my reply to Bill.
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