Apologies if my language is imprecise or poor, I'm not mathematically educated, simply a curious learner.
Working from the comments, I have edited my question for language and specificity.
I'm interested in topologies which self-intersect in n dimension. Do all such topologies not self-interect in n+1 dimension? If so, how do we know it's always true? Is there a name for this phenomenon? What happens to an n dimensional self-intersection at n-1 dimensions?
Like if I have a klein bottle which I'm viewing in three dimensions, and I added a second identical neck alongside the first one, thereby creating new self intersection points (where the two necks diverge from each other and then merge back into the bottle) these self-intersections would not occur in the fourth dimension. If this is correct, is there any way to get an intuitive picture of why it is the case? What would happen to them in the second dimension?
You may be looking for "The Whitney Embedding Theorem", which says that an $m$-dimensional smooth manifold can always be embedded in a $2m$-dimensional space, and that $2m$ is the best possible result. So your Klein bottle, which is a surface, so, a $2$-dimensional manifold, can be embedded in $4$-space, but not in any smaller dimension. See https://en.wikipedia.org/wiki/Embedding#Differential_topology and/or https://en.wikipedia.org/wiki/Whitney_embedding_theorem