Given a cardinal $\lambda > 2^{\aleph_0}$, is it possible to construct a connected $T_2$-space $(X,\tau)$ with $|X|=\lambda$?
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3I think a generalization of the long line will work. Longer versions certainly aren’t path connected but I see no reason why they wouldn’t be connected. – spaceisdarkgreen Dec 23 '19 at 15:30
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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 $2^{\aleph_0}$.
Hanul Jeon
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For a path connected solution, take $\lambda$ copies of the closed unit interval $[0,1]$ and identify the $0$s in all copies.
Pedro Sánchez Terraf
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+1... Your example is path-connected and completely metrizable..... Let $|S|=\lambda.$ The HedgeHog of Spininess $\lambda$ is ${0}\cup ((\Bbb R\setminus {0})\times S)$ with the metric $d(0,(r,s))=|r|,; d((r,s),(r',s))=|r-r'|,;$ and $d((r,s),(r',s'))=|r|+|r'|$ when $s\ne s'.$ – DanielWainfleet Dec 29 '19 at 23:15
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Thank you very much @DanielWainfleet... I suspected it had all the separation properties but I was too lazy to verify them rigorously :-). – Pedro Sánchez Terraf Dec 30 '19 at 13:12
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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).
Henno Brandsma
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