Question

space X.

X $X$.

$X$ is compact.

X^κ

$X^\kappa$ is Lindelöf for any cardinal

X^{ω_1$\kappa$.}

$X^{\omega_1}$ is Lindelöf.

for 1 implies 2, since if X $X$ is compact, thenXκ$X^\kappa$ is compact and hence Lindelöf.

So suppose that we have a space X $X$ that is not compact, butX^{ω_{1$X^{\omega_1}$ is Lindelöf. Itfollows that X $X$ is Lindelöf. Thus, there is a countableU0$U_0 \subset U1U_1 \subset ... Un... \dots U_n \dots$with the union U{ Un$\bigcup\lbrace U_n \; | \; n \in ω} \omega \rbrace = XX$.}}

For each $J \subset ω_{1\omega_1$ of size n, $n$, let UJ$U_J$ bethe set {$\lbrace s \in X^{ω1X^{\omega_1} \; | s(α)\; s(\alpha) \in UnU_n$ for each α $\alpha \in J}. J \rbrace$. As the size of J$J$ increases, the set UJ$U_J$ allows more freedom on thecoordinates in J, $J$, but restricts more coordinates. If J $J$ hassize n, $n$, let us call UJ$U_J$ an open n-box, $n$-box, since itrestricts the sequences on n $n$ coordinates. Let F $F$ be thefamily of all such UJ$U_J$ for all finite $J \subset ω1.\omega_1$}}

This F $F$ is a cover of X^ω₁. $X^{\omega_1}$. Tosee this, consider any point $s \in X^{ω_{1X^{\omega_1}$. For each α $\alpha \inω1, \omega_1$, there is some n $n$ with s(α) $s(\alpha) \inUnU_n$. Since ω1$\omega_1$ is uncountable,there must be some value of n $n$ that is repeated unboundedlyoften, in particular, some n $n$ occurs at least n $n$ times. Let Jbe the coordinates where this n $n$ appears. Thus, s $s$ is inUJ. $U_J$. So F $F$ is a cover.}}

Since X^{ω_{1$X^{\omega_1}$ is Lindelöf,there must be a countable subcover F0. $F_0$. Let J* $J^*$ bethe union of all the finite J $J$ that appear in theUJ$U_J$ in this subcover. So J* $J^*$ is a countable subsetof ω1. $\omega_1$. Note that J* $J^*$ cannot be finite,since then the sizes of the J $J$ appearing in F0$F_0$Xω1. $X^{\omega_1}$. We may rearrange indicesand assume without loss of generality that J*=ω $J^*=\omega$ isthe first ω $\omega$ many coordinates. So F0$F_0$ isreally a cover of Xω, $X^\omega$, by ignoring the}}

But this is impossible. Define a sequence $s \inX^{ω_{1X^{\omega_1}$ by choosing s(n) $s(n)$ to beoutside Un+1, $U_{n+1}$, and otherwise arbitrary. Note thats $s$ is in Un$U_n$ in fewer than n $n$ coordinates belowω, $\omega$, and so s $s$ is not in any n-box $n$-box with $J \subset ω, \omega$, since any such box has n $n$ values in Un.Thus, s $s$ is not in any set in F0, $F_0$, so it is not a}}

In particular, to answer the question at the end, it suffices to take any uncountable kappa. $\kappa$.