TITLE: Spaces without maximal homogeneous subspaces
QUESTION [8 upvotes]: A homogeneous space $(X,\tau)$ is a topological space such that for all $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x)=y$. As a previous question implies, the union of an ascending chain of homogeneous spaces need not be homogeneous (example, see below).
What is an example of a Hausdorff space $(X,\tau)$ that does not contain a homogeneous subspace that is maximal with respect to $\subseteq$?
Note. The supremum (union) of an ascending chain of homogeneous spaces need not be homogeneous: Endow $\mathbb{N}$ with the discrete topology and consider the disjoint union $X = (\mathbb{N}\times\{0\}) \cup (\mathbb{Q}\times\{1\})$, where $\mathbb{Q}$ carries the Euclidean topology. Since $X$ is countable, it is the the union of an ascending chain of finite sets, all of which carry the discrete topology and are homogeneous, but $X$ is not.

REPLY [2 votes]: A comment on Taras Banakh's beautiful answer (with his corrections). If I understand it correctly, the key ingredients of the construction are:
1) $X$ is a regular space, countable union of disjoint open sets $X_k$, for $k\in\omega$; 
2) Any nonempty open subset of $X_{k+1}$ contains a copy of $ X_k$ as a subspace.
As a consequence of 1) and 2), any open subset of $X$ that meets infinitely many $X_k$, contains a copy of $X$ itself as a subspace (it has room enough to include  a topological sum of copies $X_k$,  for all $k$); in fact even with its closure.  
3)   Each subset of $X_{k+1}$ , which is homeomorphic to a subset of $X_{k}$ is not dense in $X_{k+1}$
In this situation if  $H\subset X$ is a homogeneous subspace of $X$,   then    $ X_k\setminus \overline H\neq\emptyset$   for all large  $k $. But then  $X$ also contains a shifted copy $H'$ of $H$, separated by neighbourhoods, so that it also contains their topological sum, which is a strictly  larger homogeneous space.