A set very different from itself Let $(X,\tau)$ be a regular space (having at least two points). Let's call $X$ self-different if the only homeomorphism $\phi:X\to X$ is the identity function.
I know that you can have examples when $X$ is $T_0$. One is when $X=\{1,2\}$ and $\tau=\{\emptyset,\{1\},X\}$. I believe that if $X$ is Hausdorff it cannot be self-different, but I have no idea of how to prove that. What is the strongest separation axiom in which you can define self-different spaces?
 A: Even the real line can have a rigid subspace $X$. I can present a short construction. Also I googled an example of a rigid subspace of the real line whose square is a homogeneous subspace of the plane [L].
My construction is the following. Since the real line is second countable, that is it has a base of  cardinality $\omega$, it has at most $2^\omega$ open subsets and so at most $(2^\omega)^\omega=2^\omega$ $G_\delta$-subsets. Let $G$ be any of them. Since the real line is  second countable, it is hereditarily separable, so $G$ has a countable dense set $D$. Since the real line is Hausdorff, by [E, Th.1.5.4] a continuous map from $G$ to a real line is uniquely defined by its restriction on the set $D$. Since $|D|=\omega$, there is at most $(2^\omega)^\omega=2^\omega$ such maps. The above arguments imply that there is at most $2^\omega$ maps which are homeomorphic embeddings of subsets of the reals into it. The converse inequality is obvious, so let $\{f_\alpha:\alpha<2^\omega\}$ be an enumeration of all such embeddings. 
By transfinite induction define a sequence $\{y_\alpha\}=Y$ of points such that  $y_\alpha\in \Bbb R\setminus \{f_\beta(y_\gamma), f^{-1}_\beta(y_\gamma):\beta,\gamma<\alpha\}$. Let $Y_0$ be a set of points of $Y$ which has an open neighborhood $O_y$ of cardinality less than $2^\omega$. Let $\{O_y:y\in Y_0\}$ be a countable cover of the set $Y_0$. Since the real line is  second countable, it is hereditarily Lindelöf, so there exists a countable subset $Y_1$ of $Y_0$ such that $Y_0\subset\bigcup\{O_y:y\in Y_1\}=Z$. König’s Theorem [Ko] implies (see, for instance, [J, Cor. of Th. 9]) that $2^\omega$ has uncountable cofinality, that is it cannot be a union of a countable family of sets of smaller cardinality. Therefore $|Z|<2^\omega$, so the set $X=Y\setminus Z$ is non-empty and each its non-empty open subset has cardinality $2^\omega$.
We claim that the space $X$ is rigid. Indeed, assume the converse, there exists a non-identity homeomorphism $f'$ of the space $X$. Let $x\in X$ be a point such that $f'(x)\ne x$. Let $O_x$ and $O_{f'(x)}$ be disjoint neighborhoods of the points $x$ and $f'(x)$, respectively. Since the map $f'$ is continuous at $x$, there exists an neighborhood $O’_x$ of the point $x$ such that $f'(O’_x)\subset O_{f'(x)}$. Let $\operatorname{non-fix}(f')$ be the set of all non-fixed points of the map $f'$. Since the set $\operatorname{non-fix}(f')$ contains a non-empty open subset $O’_x \cap O_x$ of the space $X$, it has cardinality $2^\omega$. By Lavrentiev’s Theorem [Ke 3.9], $f’$ can be extended to a homeomorphism $f$ between $G_\delta$-sets of the real line. Therefore $f=f_\alpha$ for some $\alpha<2^\omega$. Let $y\in \operatorname{non-fix}(f’)$. Then $y=y_\beta$ for some $\beta<2^\omega$ and $f’(y_\beta)=f_\alpha(y_\beta)=y_\gamma$ for some $\gamma<2^\omega$. Assume that $\beta>\alpha$. Then, by the definition of the set $Y$, $\gamma\le\beta$. Since $y\in \operatorname{non-fix}(f_\alpha)$, the equality is impossible, so $\gamma<\beta$.  On the other hand, $y_\beta=f_\alpha^{-1}(y_\gamma)$, which contradicts to the definition of the set $Y$. Thus $\operatorname{non-fix}(f’) \subset\{y_\beta:f_\alpha(y_\beta)\in Y\}\subset\{y_\beta:\beta\le\alpha\}$, so $|\operatorname{non-fix}(f’)|\le \alpha<2^\omega$, a contradiction.
References
[E] Ryszard Engelking, General Topology, (2nd ed), – Heldermann, Berlin, 1989.
[J] Thomas J. Jech, Lectures in set theory with particular emphasis on the method of forcing, – Springer-Verlag, Berlin-Heidelberg-New York, 1971 (Russian translation, Mir, Moskow, 1973).
[Ke] Alexander S.  Kechris, Classical Descriptive Set Theory, – Springer, 1995.
[Ko] J. König, Zum Kontinuum-Problem, Math. Annalen,60 (1905), 177-180. 
[L] L. Brian Lawrence, A rigid subspace of the real line whose square is a homogeneous subspace of the plane, Trans. Amer. Math. Soc. 357:7 (2005), 2535-2556.
