I try to understand the notion of tubular neighborhood, but I cannot.
Is there anyone who may explain for example (up to homeomorphism) what is the tubular neighborhood of circle and disk?
For the second one, is it annulus?
2026-03-27 08:46:08.1774601168
Tubular Neighborhood of Disk and Circle
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Imagine a very thin piece of wire floating in space.
Suppose that the wire is not pinched anywhere, and does not touch itself.
Then it is possible, without moving the wire at all, to thicken it.
The thickened wire is a tubular neighbourhood (in 3-space) of the original thin piece of wire.
To your examples:
Suppose that your thin piece of wire lives in the plane (2-space) and forms a loop. Since it doesn't pinch or self intersect, you can thicken it. Since it lives in the plane, you can only thicken it the direction of the plane. When you do so, it just looks like a thicker version of the same circle.
How do you thicken something that is already a solid. For instance, a disc in the plane (2-space). Imagine thickening the boundary (which is a loop, described in the previous paragraph) and then adding the result to the original disc. The result is a tubular neighbourhood of the disc.
Note that in the examples, the tubular neighbourhood depends on the ambient space: a tubular neighbourhood of a loop in the plane is different (homeomorphically) from a tubular neighbourhood of a loop in 3-space.