I don't understand visually or analytically why the three images with arrows are not manifolds.
It would be nice to have an intuitive explanation too why they are not manifolds, similar to this explanation for what is a manifold:
"So a manifold is something that, looking at a small piece, always looks like a some linear space, of a consistent dimension. A submanifold is just a subset of a manifold that is itself a manifold (of smaller dimension). An example would be the equator as a submanifold of the surface of earth. The equator is a giant circle, like the big circle of wire above. It's a subset of the earth's surface, so it's a submanifold of it."
And then, how can we prove that the square, the line are manifolds?
Definition of homeomorphic: two topological spaces X and Y are homeomorphic if $\exists f: X \rightarrow Y$ s.t. f is bijective, f is continuous, and $f^-1$ is continuous.
I have some thoughts, please judge if they are correct. I will refer to the figures by number 1-8, with the top left rectangle being 1, the top right bow being 4, and the bottom right T-shape being 8.
Mainly, I understand homeomorphism in the above examples to be something like, if we zoom in as much as possible on each point and think about what the infinitesimal neighborhood around that point looks like, does it look like a line ($R^1$) or plane ($R^2$)?
Figure 1: I assume the manifold here refers to the points along the edges of the rectangle and not the inner area itself. The points along the edge of the rectangle are homeomorphic to $R^1$ and the corner points are homeomorphic to $R^2$. But actually, the corner points don't seem homeomorphic to $R^2$ here if the center point of Figure 4 (the bow) is not homeomorphic to $R^2$ - in both cases, there is not a radial neighborhood of valid points around the target point (corner in rectangle and center in the bow) in question.