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:


sphere with two holes

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

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    $\begingroup$ Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges. $\endgroup$ Aug 26, 2018 at 1:27
  • 4
    $\begingroup$ 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. $\endgroup$
    – user247327
    Aug 26, 2018 at 3:31
  • 1
    $\begingroup$ The drawing is perhaps not the best but hopefully the others explained it. $\endgroup$
    – Tom
    Aug 27, 2018 at 9:50

4 Answers 4


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.

(left pic is from post, right pic is super contrast enhanced to tease out line)

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

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    $\begingroup$ I see the trench, but which one is the superlaser and which one is the exhaust port? $\endgroup$ Aug 26, 2018 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.)

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    $\begingroup$ Or you inflated the flattened donut, which happens to have a weak, more redundant part. $\endgroup$ Aug 25, 2018 at 19:07
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    $\begingroup$ 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. $\endgroup$
    – Fogmeister
    Aug 26, 2018 at 6:47

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

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    $\begingroup$ 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. $\endgroup$ Aug 25, 2018 at 20:28
  • $\begingroup$ @AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too. $\endgroup$
    – dmtri
    Aug 26, 2018 at 6:10

enter image description here 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.


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