Two disjoint circles in the Euclidean space are homeomorphic to two circles interlocked without touching each other. My professor said that to a topologist they are the same thing. I don't understand why. What properties of a space are of concern to a topologist and what are things he is not interested in?
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They are the "same thing" in the sense that there is a homeomorphism between them. They look different to you because you are considering how the two components are embedded in $\mathbb R^3$. The circles don't "know" about the embedding in the ambient space you see.
There are other ways in which they are not the same thing, but those involve more than just comparing them as topological spaces. You know intuitively that in some way there is a difference (otherwise, handcuffs wouldn't work), but he is speaking from a purely topological perspective.
Homeomorphisms capture the intrinsic properties of a topological space.
The distinction you want to make is about extrinsic properties: how the two circles are embedded in euclidean space. One way to phrase this precisely is by considering the complement in the two cases. These complements are not homeomorphic.
Check out link theory.