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 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).

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