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Let $X$ be a non-empty and topological space. The set of all real-valued continuous function and all non-constant real-valued continuous function on $X$ is denoted by $C(X)$ and $NC(X)$, respectively.

Clearly, $C(X)\not=\emptyset$ since every constant function on $X$ is continuous. Also, if $|X|=1$, then $NC(X)=\emptyset$.

Does there exists a topological space $X$ with $|X|>1$ and non-trivial, which $NC(X)=\emptyset$. In the other words, does there exists a topological space $X$ with $|X|>1$ such that every real-valued continuou functions on $X$ is constant?

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  • $\begingroup$ In General Topology by Engelking there is given an example of an infinite space which is $T_1$ and $T_3$ and on which every continuous real-valued function is constant. I may try to find it in my copy. $\endgroup$ – DanielWainfleet Dec 20 '16 at 19:21
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    $\begingroup$ In this answer I describe a construction, due to Eric van Douwen, that starts with any $T_3$ space having two points that cannot be separated by a continuous real-valued function and produces a $T_3$ space on which every real-valued continuous function is constant. In the highlighted section of this answer I describe a relatively simple example, dut to John Thomas, of a $T_3$ space with two points that cannot be separated by a continuous real-valued function. $\endgroup$ – Brian M. Scott Dec 21 '16 at 21:51
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Another non-trivial example:

Let $X$ be an infinite set and $\mathcal T$ the co-finite topology on $X$. Then $C(X)$ consists of only constant functions.

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    $\begingroup$ +1. Note that what's really going on here is: if $X$ is a topological space with no disjoint nonempty open sets, then $X$ has no nonconstant continuous maps to any Hausdorff space. This may seem like a very restrictive condition, but there are natural spaces which have it - in particular, the Zariski topologies. $\endgroup$ – Noah Schweber Dec 20 '16 at 20:55
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Sure, take any space $X$ endowed with the trivial topology.

Edit: There are also nontrivial examples, for example consider $\{\emptyset, A, X\}$ for any arbitrary subset $A \subset X$.

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  • $\begingroup$ Thx. The question is edited. $\endgroup$ – M.A. Dec 20 '16 at 18:27
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    $\begingroup$ I have extended my answer. Please refrain from changing questions too much, as this invalidates previous answers and generates confusion. $\endgroup$ – Dominik Dec 20 '16 at 18:30
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You can see this paper by Edwin Hewitt, where it is proved that for every cardinal number $\aleph$ which is not the sum of $\aleph_0$ cardinal numbers less than $\aleph$, there exists a Hausdorff space $X$ with $|X|=\aleph$ such that every continuous real function on $X$ is constant.

Some examples by Urysohn and Pospišil are mentioned.

Theorem 2 in the paper by Hewitt exhibits a countable Hausdorff space on which every continuous real function is constant. The construction is due to R. F. Arens.

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  1. Edwin Hewitt, On two problems of Urysohn, Annals of Mathematics Second Series, Vol. 47, No. 3 (Jul., 1946), pp. 503–509

  2. Paul Urysohn, Über die Mächtigkeit der zusammenhängenden Mengen, Mathematische Annalen, 94 (1925), 262–295

  3. Bedřich Pospišil, Trois notes sur les espaces abstraites, Spisy vidáváne přirodoveckou Fakultou Masarykovy Univerzity, Čis. 249, 1937

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