Large connected $T_2$-spaces Given a cardinal $\lambda > 2^{\aleph_0}$, is it possible to construct a connected $T_2$-space $(X,\tau)$ with $|X|=\lambda$?
 A: The idea of @spaceisdarkgreen works. 
In my previous answer, I showed that $\mu\times [0,1)$ is a linear continuum (that is, a linear order which is dense and complete) for a limit ordinal $\mu$. Therefore,
if $\kappa>2^{\aleph_0}$, then $\kappa\times [0,1)$ is a linear continuum, which is connected and has cardinality $\kappa$.
Note that every linear order is $T_5$, so we actually have a connected $T_5$ space of an given cardinality greater than $2^{\aleph_0}$.
A: For a path connected solution, take $\lambda$ copies of the closed unit interval $[0,1]$ and identify the $0$s in all copies.
A: Let $X$ be the subspace of $[0,1]^\lambda$ (itself a connected Hausdorff space of size $2^\lambda > \lambda$) of functions of finite support (i.e. $\{\alpha \in \lambda: f(\alpha) \neq 0\}$ is finite). This is is easily seen to be (even path-)connected as (essentially a) union of cubes $[0,1]^n, n \in \omega$ all with the constant $0$ function in their intersection. $X$ has size $\lambda$ (as a function in $X$ is determined by a finite choice from the domain $\lambda$ (support) followed by $\mathfrak{c}< \lambda$ choices for the values).
A: $\omega_{\beta} × [0,1)$ in lexicographical order.
