I’ve been thinking about loneliness as a topological space.
Consider an open set U containing a single point: you.
In this space, every neighborhood of you still fails to cover the entire set of “companionship” you imagine. The closure of these sets is always incomplete; there’s always a gap, an epsilon of emptiness that no limit point can approximate.
We often talk about “connectedness” in life, in a social sense. But what if the space of our minds is fundamentally disconnected, or worse, totally non-Hausdorff? You try to separate points (your past self from your present, your ideal self from the version they see) but no disjoint open sets exist. Everything bleeds together like ink on damp paper.
I wonder if our desire for intimacy is an attempt at constructing a compactification. We try to add points at infinity - stories, relationships, late-night confessions - hoping they’ll make the space “nice”, or at least easier to navigate. But perhaps the one-point compactification of the soul is too simplistic. Maybe we need an entire Stone–Čech compactification, a kind of hyper-extension of our emotional universe, containing every ultrafilter of what we might have been.
You might think: “this is all unnecessarily abstract. loneliness is simply a lack of people.” But is it? Or is it the inability to construct a continuous map from the space of your mind to the space of another’s? A failure of surjectivity! An unsolvable preimage problem!!
I keep trying to define a metric on my internal landscape , hoping to measure closeness, to optimise paths. But perhaps loneliness is fundamentally non-metrisable. Perhaps it’s a fractal boundary we keep tracing with trembling hands, over and over..
Some loose ends:
- Can we define a homotopy between two states of being: alone and together?
- Is there a fundamental group of loneliness?
- What would a continuous deformation of solitude into connection look like?