How can there exist a quotient map to a space of higher dimension than the domain? I don't need a completely formal explanation, just some intuition. My professor stated
identifying antipodal points on $S^1$, yields the projective plane $P^2$. That means there exists a quotient map (and thus a surjective map) $q: S^1\to P^2$.
Intuitively, this confuses me. How can identifying points together raise the dimension of the resulting space? Lowering or keeping the dimension both make sense to me, but I can't wrap my mind around it raising the dimension. Like, I knew that surjective linear maps $T: X\to Y$ require that $\dim Y \leq \dim X $, but this clearly doesn't hold for all continuous maps, just linear ones.
Could someone offer some intuition behind this, or how they think about it? I can wrap my head around the codomain being "larger" than the domain (like a continuous bijection $f: [0, 1]\to\mathbb{R}$, even though since there's a bijection $\mathbb{R}$ technically isn't larger) but not the codomain having higher dimension.

It’s just an error, by you or by your professor. The projective plane results from identifying antipodal points on $S^2.$
However, there do exist counterintuitive quotient maps that increase dimension. A space-filling curve is a continuous surjection $[0.1]\to [0,1]^2$; since the domain is compact and the codomain is Hausdorff, this is also a closed map, thus in particular, a quotient map. (This cannot happen with differentiable functions, or with injective continuous functions, happily.)
