My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.


The *generalized Cantor space* is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.


A space is *$\kappa$-compact* if every open cover has a subcover of cardinality (strictly) smaller than $\kappa$.


For which cardinals $\kappa$ is the generalized Cantor space $2^\kappa$ $\kappa$-compact? We are interested in strong limit cardinals $\kappa$, mainly strongly inaccessible $\kappa$.


We would especially appreciate references.


Update: [A related question.](https://mathoverflow.net/questions/234926/when-does-the-generalized-cantor-space-embed-in-a-kappa-compact-space)



When is the generalized Cantor space $\kappa$-compact?
A cardinal $\kappa$ is weakly compact if and only if $2^{\kappa}$ is $\kappa$-compact and $\kappa$ is strongly compact if and only if $2^{I}$ is $\kappa$-compact for all sets $I$ where $2^{I}$ is given the topology with basis of open sets of the form $[\sigma]$ where $\sigma:J\rightarrow 2$ and $|J|<\kappa$.


To prove the tree property from the Tychonoff theorem characterization of weak compactness, one uses the following argument. Suppose that $T$ is a $\kappa$-tree. Then for each $\alpha<\kappa$, let $T\_{\alpha}$ be the $\alpha$-th level of this tree. Then by a compactness argument, one can show that the inverse limit $\varprojlim\_{\alpha<\kappa}T\_{\alpha}\subseteq\prod\_{\alpha<\kappa}T\_{\alpha}$ is a non-empty closed subset (the proof of this fact is the same in the ordinary compactness case). However, $\varprojlim\_{\alpha<\kappa}T\_{\alpha}$ is the set of all $\kappa$-branches of $T$. Therefore $\kappa$ satisfies the tree property.


To prove that every weakly compact cardinal satisfies the $\kappa$-Tychonoff theorem, one could use the tree property and inaccessibility to show that every $\kappa$-filter on a $\kappa$-algebra of sets can be extended to a $\kappa$-ultrafilter and from this fact one could show that weakly compact cardinals satisfy a $\kappa$-version of Alexander's subbase theorem. From Alexander's subbase theorem, one then establishes that weakly compact cardinals satisfy the $\kappa$-Tychonoff theorem.


This result was originally proven in the paper [Additions to some Results of Erdos and Tarski](http://matwbn.icm.edu.pl/ksiazki/fm/fm53/fm53124.pdf) by Donald Monk and Dana Scott.


Furthermore, a long list of characterizations of weak compactness including several topological characterizations is given in the book The Theory of Ultrafilters (1974).


