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From: William Elliot on 27 Jul 2010 00:13
Let S be a set and p an element of S.

What topology can S have if for all functions f:S -> S,
f is continuous iff f(p) = p or f is constant function?

Is the topology unique? If it's required that the space
be T0, will the topology be unique?
From: W. Dale Hall on 27 Jul 2010 01:25
William Elliot wrote:
> Let S be a set and p an element of S.
>
> What topology can S have if for all functions f:S -> S,
> f is continuous iff f(p) = p or f is constant function?
>
> Is the topology unique? If it's required that the space
> be T0, will the topology be unique?

You might want to note that the identity self-map is
continuous for any topology.
From: William Elliot on 27 Jul 2010 02:41
On Mon, 26 Jul 2010, W. Dale Hall wrote:
> William Elliot wrote:

>> Let S be a set and p an element of S.
>>
>> What topology can S have if for all functions f:S -> S,
>> f is continuous iff f(p) = p or f is constant function?
>>
>> Is the topology unique? If it's required that the space
>> be T0, will the topology be unique?
>
> You might want to note that the identity self-map is
> continuous for any topology.
>
That's one of the maps with f(p) = p.
From: W. Dale Hall on 27 Jul 2010 04:08
William Elliot wrote:
> On Mon, 26 Jul 2010, W. Dale Hall wrote:
>> William Elliot wrote:
>
>>> Let S be a set and p an element of S.
>>>
>>> What topology can S have if for all functions f:S -> S,
>>> f is continuous iff f(p) = p or f is constant function?
>>>
>>> Is the topology unique? If it's required that the space
>>> be T0, will the topology be unique?
>>
>> You might want to note that the identity self-map is
>> continuous for any topology.
>>
> That's one of the maps with f(p) = p.

oops, right. I misread your condition.
From: Butch Malahide on 27 Jul 2010 04:14
On Jul 26, 11:13 pm, William Elliot <ma...(a)rdrop.remove.com> wrote:
> Let S be a set and p an element of S.
>
> What topology can S have if for all functions f:S -> S,
> f is continuous iff f(p) = p or f is constant function?
>
> Is the topology unique?  If it's required that the space
> be T0, will the topology be unique?

Given S and p, the only topologies satisfying your condition are:
(I) the open sets are the empty set and all sets containing p;
(II) the open sets are S and all sets not containing p.

These are T_0 topologies. If |S| = 1 they coincide; if |S| = 2 they
are distinct but homeomorphic; if |S| > 2 they are nonhomeomorphic.
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