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Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces.

I have seen some example which uses $X$ to be non sequential space but I'm interested in an algebraic example. (I mean in which space $X$ is some algebraic object like ring with Zariski topology or any other topology)

Martin Sleziak
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Arpit Kansal
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  • The example I've seen is taking $X = \omega_1 + 1$ where $\omega_1$ is the first uncountable ordinal and $Y = \Bbb R$. You then set $$f(\alpha) = \cases{0 & if $\alpha < \omega_1$\ 1 & if $\alpha = \omega_1$}$$This is a sequentially continuous function, but not a continuous one, if $X$ is endowed with the order topology and $Y$ with the standard one. – Arthur Aug 28 '15 at 10:51
  • I will add a few links to other post on this site which have examples of such functions: http://math.stackexchange.com/q/351987, http://math.stackexchange.com/q/745036 and perhaps also http://math.stackexchange.com/questions/53236 (this is a slightly different question). – Martin Sleziak Aug 28 '15 at 12:37
  • Could you be more specific what you mean by algebraic example. The current wording of the question might lead to the impression that you only allow Zariski topology for $X$. (In which case the question can be solved by finding an example of a ring for which the Zariski topology is not sequential. – Martin Sleziak Aug 28 '15 at 12:41
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    Some remarks about the general approach to this kind of problems: Once you found a space $X$ which is not sequential, this means precisely that there is some map $f:X\to Y$ such that $f$ is sequentially continuous but not continuous. But finding the space $Y$ and the map $f$ is not even a problem, as you can simply take $f=\text{id}:X\to sX$, where $s$ is the sequentiallization of $X$, i.e. the set $X$ equipped with the topology consisting of all sequentially open subsets of $X$. – Stefan Hamcke Aug 28 '15 at 13:33
  • In fact, this sequentially continuous map $X\to sX$ is universal from $X$ to the inclusion functor of sequential spaces into the category of all spaces with sequential maps as arrows. So in the end, the problem boils down to finding a non-sequential space. – Stefan Hamcke Aug 28 '15 at 13:33
  • @StefanHamcke Thank you.Regards, – Arpit Kansal Aug 28 '15 at 15:32
  • @MartinSleziak I'm unable in finding a ring for which Zariski topology works here?How do you find this ring?[Actually I don't have much idea about topologies (except Zariski Topology on Ring)on algebraic objects like Groups,rings,Vector Spaces etc...Can you say some lines about these topologies?]Best Regards, – Arpit Kansal Aug 28 '15 at 15:36
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    @ArpitKansal I am not very familiar with the Zariski topology. However answer to this question make me believe that working with sequences in Zariski topology will not be very easy. – Martin Sleziak Aug 31 '15 at 08:07
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    possible duplicate of Looking for example of topological spaces where sequential continuity does not imply continuity – Nate Eldredge Sep 20 '15 at 17:41

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