This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is $\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked whether $\mathbb{R}^3$ remains simply connected after deleting a countable set of points, such as the collection of rational points $\mathbb{Q}^3$.

That question was answered affirmatively by Martin M. W. My question is, can we do better? Specifically, I want to understand, in general context where the continuum may be very large, exactly how many points we may freely delete from $\mathbb{R}^3$, whilst remaining simply connected. What is the fewest number of points that we must delete from $\mathbb{R}^3$ in order to make it no longer simply connected?

Let us define the *simply
connected deletion number*, $\delta$, to be the smallest cardinality of a subset
$A\subset\mathbb{R}^3$, such that the complement
$\mathbb{R}^3\setminus A$ is no longer simply connected.

Martin's answer to the earlier question shows that deleting any countable number of points preserves the simply connected property, and so the simply connected deletion number is definitely uncountable, at least $\omega_1$. And since it is clearly at most the continuum, the question is settled if the continuum hypothesis holds. Like all other cardinal characteristics of the continuum, this number is more interesting when the continuum hypothesis fails.

In a comment,
I had suggested that Martin's argument suggested that the simply
connected deletion number should be at least as large as
**cov**$(\cal{M})$, the covering number of the meager
ideal, which is the fewest number of meager sets whose union is
the whole space. My reason for suggesting this was that as far as I understand
Martin's answer (which I admit is imperfectly), he is proposing that for
any one point $x$, there is a comeager set of homotopies that
avoids $x$. So in order to avoid all the points in a set $P$, we need to
know that the intersection of $|P|$ many comeager sets in his space of homotopies is
nonempty. This is the same as knowing that the unions of $|P|$
many meager sets (the complements) is not the whole space of homotopies, in order that there is at
least one desired homotopy that avoids every point in $P$.

If this is right, then we would deduce that the simply connected
deletion number is at least **cov**$(\cal{M})$, provided that the covering number for meager sets in his space was the same as for our other more familiar spaces. (If someone
could explain and confirm this inequality in greater detail, please post
an answer! I would want to see more details than Martin had provided about the space of homotopies.)

**Question.** What is the simply connected deletion number
exactly? Is it consistent that this number is strictly less than
the continuum? Is it necessarily the continuum? How does it relate to the other standard cardinal
characteristics of the continuum? What is the value under Martin's
axiom? Is it equal to **cov**($\cal{M}$)? Can it be
strictly larger than **cov**$(\cal{M})$?