Can one imagine "a point" as counter to Kant's "space mind experiment?" Yes and No :)

1. In geometry: a point has zero extension, but it is not nothing

For example: In Euclidean geometry, a point is an object with position but no length, area, or volume.

So, in one dimension:

$$\text{length of a point}=0$$

That doesn't mean that the point is identical to the number $0$.

For example, imagine on a line, the points

$$-5,\quad 0,\quad 7$$

all have zero length. Yet they are different points because they occupy different positions.

So:

zero is a measure; a point is a location. 

2. In topology, the distinction becomes even clearer:

Topology doesn't fundamentally care about length, area, or volume. 

It cares about neighborhoods, continuity, connectedness, and how points relate to one another.

A topological point is an element of a space:

$$p\in X.$$

The point $p$ can have no size whatsoever and still be structurally significant.

For example, consider the interval

$$[0,1].$$

The point $0$ is a single point. Its measure is zero. But topologically it isn't equivalent to “nothing”: removing its left bound it gives

$$(0,1],$$

which is a different topological space.

Even more strikingly, removing a single point from the circle $S^1$ produces something homeomorphic to a line (sorry I cannot reproduce a drawing here), the result would be:

$$$$S1−{p}≅R.S^1-\{p\}\cong \mathbb R.

So a point can have zero size while having important structural consequences.

3. Is a point therefore “no space”?

Here we need to be careful.

A point is zero-dimensional, but that doesn't mean it is “outside space.”

A point is an element of a space.

Think of:

$$\mathbb R$$

as the space and

$$3$$

as one of its points.

The point 3 doesn't contain any interval around itself. But the space does. Every “neighborhood” of 3 contains other points:

$$(2.9,3.1).$$

This gives us an interesting distinction:

The point itself has no extension; but its neighborhood expresses its spatial situation.

Why is that?

Because dimension is not the same thing as size. 

A point has zero size in every direction, so its dimension is 0.

There are two ways to understand this: geometrically and topologically, and the topological one is more interesting.

Geometrically

A line has one independent direction:

$$\longrightarrow$$

You can move along it in one dimension. A plane has two:

$$\rightarrow \quad \uparrow$$

A point has neither. You can't move in any direction while staying within the point.

So: 

point → 0 dimensions; 

line → 1 dimension; 

plane → 2 dimensions; 

ordinary space → 3 dimensions;

This distinction becomes very important philosophically.

4. There's a beautiful distinction coming up:

Let's come back to our Kantian experiment.

 ¿Can we imagine a room without objects, but cannot imagine the room without space?

Yes! 

You can imagine a point without extension:

$$\bullet$$

but you cannot really get a spatial point without some spatial structure in which “point” has its meaning.**

However, 

A point has no space of its own. That is to say: it cannot function as a point except within a space.

That's different from saying that a point is “no space.”

___________________

** A point has zero extension, but it nevertheless has position.

Where does that position come from?

Not from the point itself.

If you opened up the point and looked inside, you'd find no spatial content explaining its position. Its position is determined ONLY relationally:

It is here rather than there.

And "here rather than there" presupposes a field of possible locations.

It doesn't mean that you need to physically draw a line around the point. 

The concept of a spatial point is incomplete without the possibility of spatial relations.