Suppose we have as a topological space $X$ the figure eight space. Let $A \subset X$ be one of the circles, then evidently one has a continuous map $f: X \rightarrow A$ by mapping the other circle to the point of intersection. Often, when I visualize continuous functions, I consider them as if they are "deformations" of the space (I do NOT mean the real topological notion of 'deformation', but the one you find in the dictionary).
However, this has some complications: for this to make sense (as highlighted by my example) I would always need some ambient space in which I can visualise this deformation. In my case, this is evident: just use the Euclidean space, then the 'continuous deformation' is the one collapsing the circle to said point.
However, I don't feel comfortable with just embedding everything into the Euclidean space, or some other bigger ambient topological space. My question is then: is this intuition bad? If yes, what is a better intuition to know if a map is 'continuous'? (I mean this broadly, I know that we can't always embed it into an Euclidean space, but this doesn't take away the fact that I can look at continuous maps as 'deformations' of the topological space)
Topology is not about form, geometry is about form, topology is a way to deform the properties of the space, not the space itself, at least in my point of view.