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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 $\kappa$.

Problem. For which cardinals $\kappa$ does the generalized Cantor space $2^\kappa$ embed as a subspace of every $\kappa$-compact set $C\subseteq 2^\kappa$ with $|C|>\kappa$?

It is a classic result that this holds for $\kappa=\omega$.

In the original formulation of this problem, I mentioned that, in light of an earlier answer, the case where $\kappa$ is weakly compact is particularly interesting, and the expected answer is "for all of these". According to Yair Hayut's answer below, this is wrong. The full problem, as stated above, remains open (we need cardinals for which the answer is positive).

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 $\kappa$.

Problem. For which cardinals $\kappa$ does the generalized Cantor space $2^\kappa$ embed as a subspace of every $\kappa$-compact set $C\subseteq 2^\kappa$ with $|C|>\kappa$?

It is a classic result that this holds for $\kappa=\omega$.

In the original formulation of this problem, I mentioned that, in light of an earlier answer, the case where $\kappa$ is weakly compact is particularly interesting, and the expected answer is "for all of these". According to Yair Hayut's answer below, this is wrong. The full problem, as stated above, remains open (we need cardinals for which the answer is positive).

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 $\kappa$ does the generalized Cantor space $2^\kappa$ embed as a subspace of every $\kappa$-compact set $C\subseteq 2^\kappa$ with $|C|>\kappa$?

It is a classic result that this holds for $\kappa=\omega$. We (my colleagues and I) are particularly interested, in light of an earlier answer, in the case where $\kappa$ is weakly compact, and the expected answer is "for all of these". Even if so, we would appreciate 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 $\kappa$.

Problem. For which cardinals $\kappa$ does the generalized Cantor space $2^\kappa$ embed as a subspace of every $\kappa$-compact set $C\subseteq 2^\kappa$ with $|C|>\kappa$?

It is a classic result that this holds for $\kappa=\omega$.

In the original formulation of this problem, I mentioned that, in light of an earlier answer, the case where $\kappa$ is weakly compact is particularly interesting, and the expected answer is "for all of these". According to Yair Hayut's answer below, this is wrong. The full problem, as stated above, remains open (we need cardinals for which the answer is positive).

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 $\kappa$ does the generalized Cantor space $2^\kappa$ embed as a subspace of every $\kappa$-compact subset of $2^\kappa$?

It is a classic result that this holds for $\kappa=\omega$. We (my colleagues and I) are particularly interested, in light of an earlier answer, in the case where $\kappa$ is weakly compact, and the expected answer is "for all of these". Even if so, we would appreciate 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 $\kappa$ does the generalized Cantor space $2^\kappa$ embed as a subspace of every $\kappa$-compact set $C\subseteq 2^\kappa$ with $|C|>\kappa$?

It is a classic result that this holds for $\kappa=\omega$. We (my colleagues and I) are particularly interested, in light of an earlier answer, in the case where $\kappa$ is weakly compact, and the expected answer is "for all of these". Even if so, we would appreciate references.