TITLE: How far is  ﻿Lindelöf  from compactness?
QUESTION [36 upvotes]: A while ago I heard of  a nice characterization of compactness but I have never seen a written source  of it,  so I'm starting to doubt  it.  I'm looking  for a reference, or counterexample,  for the following .  Let $X$ be a Hausdorff topological space.  Then, $X$ is compact if and only if $X^{\kappa}$ is ﻿Lindelöf  for any cardinal $\kappa$.  
If the above is indeed a fact, can one restrict the class of $\kappa$'s for which the characterization is still valid? 
Note:  Here I'm thinking under ZFC.

REPLY [6 votes]: Here is a very surprising fact I was completely unaware of until yesterday, when I found it in Herrlich´s book "Axiom of Choice". It shows that the answer to the question in the title could be "very very close":

There are models of $ZF$ in which for every $T_1$ space $X$, $X$ is Lindelöf if and only if $X$ is compact.

For instance this holds in what is called Cohen first model. Also in this model, Tychonoff´s theorem holds for Hausdorff spaces, so arbitrary products of Lindelöf Hausdorff spaces are Lindelöf.