I am looking video lectures of F. Schuller about space-time geometry, in particular about manifolds. In it, right at the beginning after introducing manifolds he gives a non-example, i.e. he says that
is not a manifold as the point where both circles meet (which I made thicker to show it), it is not possible to assign a dimension in concordence with the other points. Intuitively, if I think of this set equipped with the subspace topology derived from the standard topology on $\mathbb R^2$ I am somehow convinced that every open neighbourhood around the problematic point does not resemble an open subset of $\mathbb R$ (despite I do not see how to prove this more formally). Then later in the lecture, after introducing fibres, he gives the following example
supposing this is a manifold (the circles "shrink down" to a single point, which I made thicker to highlight it, and then "expand" in lines again), indeed by definition of fibres the entire figure is what he called the total space, which by definition is a manifold.
But comparing this with the non-example, this has in some sense analog deficiencies, the line and the circle (or the vertical lines) meet in a single point, which could be seen as problematic with the same arguments as for the non-example (similar I guess the intersection of two distinct lines is not a manifold, as the intersection point is also problematic).
So, why is the second figure a manifold, but the first not, if the second also has these problematic points where the parts meet at single points?
No, in this case the total space is not a manifold, even though the base space and the fibres over any point are.