Recovering a topological space from its connected components Let $X$ be a topological space and denote by $\pi_0(X)$ its space of connected components i.e. the set of connected components of $X$ with the quotient topology induced by the topology of $X$. Note that $\pi_0(X)$ is discrete if and only if every connected component of $X$ is open in $X$ (this is always the case if $X$ has only a finite number of connected components but not true in general).
My question is : can $X$ be reconstructed from the set of its connected components (view as topological spaces with the induced topology) and $\pi_0(X)$ as a topological space.
Here are two example where it is possible : $\pi_0(X)$ is discrete because in this case $X$ is just the disjoint union of its connected components, $X$ is totally disconected because in this case $X$ is homemorphic to $\pi_0(X)$.

The answer to this question is no. Consider two subspaces of $\mathbb R^2$, $X, Y$ with
$$
X = \left(\bigcup_{n \in \mathbb N} [0,1] \times \{\tfrac1n\}\right) \cup \{(0,0)\}
$$
and
$$
Y = \left(\bigcup_{n \in \mathbb N} [0,\tfrac1n] \times \{\tfrac1n\}\right) \cup \{(0,0)\}.
$$
In both cases, the pathconnected components are infinitely many intervals and a single point. Also, in both cases, the topology on the space of connected components is that of $\{1,\tfrac12,\tfrac13,\ldots,0\}$. But these spaces are not homeomorphic, as there is an open set around the origin in $X$ which does not contain an entire other pathcomponent, but the same thing is not true for $Y$.
