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The strength of topology seems to lie in the ability to consider geometric objects without having to deal with something as obnoxious as an ambient space. However, despite reading many books and articles on the subject, I am unable to think of a simple concrete example regarding the subject; every embedding of a sphere that I devise always references R^3.

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  • $\begingroup$ Note that if you're talking topologically, you won't be able to distinguish a sphere from many other kinds of surfaces, unless you have something more than just a topology. $\endgroup$
    – pjs36
    Dec 19, 2015 at 0:36
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    $\begingroup$ Just to be clear, the definition of the 2-sphere is the set of points in $\mathbb{R}^3$ at distance $1$ from the origin. Many mathematical objects are first defined as subobjects of other objects, and topology is no different. $\endgroup$
    – Lee Mosher
    Dec 19, 2015 at 0:36
  • $\begingroup$ @Tomislav Ostojich, what's wrong with ambient space? How else can one define tangent/cotangent vector spaces? Also the Gaussian curvature is an implicit invariant, the sphere's curvature need no ambient space and is already distinguishable from that of a plane. $\endgroup$ Dec 25, 2015 at 3:18
  • $\begingroup$ @marshalcraft see: math.stackexchange.com/questions/26551/why-abstract-manifolds $\endgroup$ Sep 13, 2016 at 2:38

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The 2-sphere is the unique (up to homeomorphism) compact, connected, simply connected 2-manifold.

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If you accept the unit interval $I=[0,1]\subseteq \Bbb R$, the sphere can be constructed from $I\times I$ by identifying all points along the boundary: $$ S^2\simeq I\times I/(I\times \{0,1\}\cup \{0,1\}\times I) $$ It is also the one-point compaction of the plane, or the smash product of the circle with itself (where the circle can be defined as the one-point compaction of or a quotient space of $\Bbb R$), or a CW-complex. There are several ways to define the sphere, but most of them do implicitly use some ambient space as part of its construction, simply because euclidean space is so well known and familiar.

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No topologist am I, but the two-sphere is the universal cover of the real projective plane, and the latter certainly has an abstract definition. If you don't like that, I guess you can take the stereographic projections from two points of the sphere to get an atlas of two charts, with the gluing functions cooked up from the two stereographies. Best of all might be to call in basic Complex Variable, and cover the sphere by two planes, each a copy of $\Bbb C$, and identify a nonzero point $z$ of the one to the point $1/z$ of the other.

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