# How is possible that those shapes are equivalent in topology?

I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:

are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?

• Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges. – Michael Hardy Aug 26 '18 at 1:27
• It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid. – user247327 Aug 26 '18 at 3:31
• The drawing is perhaps not the best but hopefully the others explained it. – Tom Aug 27 '18 at 9:50

Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.

That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.

• I see the trench, but which one is the superlaser and which one is the exhaust port? – Lamar Latrell Aug 26 '18 at 4:39

The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)

• Or you inflated the flattened donut, which happens to have a weak, more redundant part. – Antoni Parellada Aug 25 '18 at 19:07
• It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology. – Fogmeister Aug 26 '18 at 6:47

You may also notice the tunel, which I agree with you it is not clear in this photo.

• I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible. – Andreas Blass Aug 25 '18 at 20:28
• @AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too. – dmtri Aug 26 '18 at 6:10

The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.