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If $f:\mathbb{R}\to\mathbb{Q}$ is continuous, then it is constant. Are there infinite connected $T_2$-spaces $X,Y$ such that the only continuous maps $f:X\to Y$ are the constant maps?

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  • $\begingroup$ Is a $T_2$ space just an obscure (to me) way of saying a Hausdorff space? $\endgroup$
    – Kevin Buzzard
    Commented Jan 15, 2017 at 16:21
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    $\begingroup$ There are plenty of examples of compact connected metric spaces $Y$ (classical or hand-made) for which only continuous maps of $I$ into $Y$ are constant. More fun would be to get also compact connected metric spaces $X$ such that the only continuous maps from one space to another, and in the other direction, are all constant. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jan 15, 2017 at 22:35
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    $\begingroup$ Dominic, do you have a copy of Counterexamples in Topology in your library? If not, then it would seem (based on the general trend of your questions) to be a valuable book to acquire. It's published by Dover, so pretty inexpensive (from where I stand, less than 10 US dollars). $\endgroup$
    – Todd Trimble
    Commented Jan 16, 2017 at 0:54
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    $\begingroup$ Incidentally, I think Włodzimierz in his first sentence might be referring to pseudo-arcs, which are kind of a canonical example (well worth knowing about). They are strange critters, but strangely commonplace in a mathematical sense (the same way everywhere nondifferentiable continuous curves are commonplace). A classical reference is by Bing: projecteuclid.org/euclid.pjm/1102613150 $\endgroup$
    – Todd Trimble
    Commented Jan 16, 2017 at 1:19
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    $\begingroup$ I will add to @ToddTrimble's comment that most a lot of data from the book Counterexamples in Topology is also included in pi-base, which is an online database of topological spaces and their properties. For various combinations of topological properties it is very easy to search for the spaces in the database which have them. (Although I am not sure whether it would actually help with this particular question.) $\endgroup$
    – Martin Sleziak
    Commented Jan 16, 2017 at 6:06

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Any real-valued function on the positive integers with the prime integer topology (subbasis of sets of the form $U_p(b)=\{b+np:n\in \mathbb{Z}, p\nmid b\}$) is constant. This is on page 82 of Counterexamples in Topology; item 4 shows that this topology is $T_2$ and item $7$ both constancy of real-valued functions and connectedness.

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Another example: let $X=[0,1]$ and let $Y\subseteq\mathbb R^2$ be the Knaster-Kuratowski fan. Both are connected and Hausdorff, but the path components of $Y$ are points, so any map $f:X\to Y$ must be constant.

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Generalizing Dejan Govc's answer...

In this paper http://www.ams.org/journals/proc/1972-032-02/S0002-9939-1972-0296913-7/S0002-9939-1972-0296913-7.pdf it is shown that every continuous function from a connected and locally connected space into a connected space with a dispersion point is constant. It's easy to prove actually.

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