MHB R^2 accumulation and open/closed

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All points $(x,y)$ such that $x^2 - y^2 < 1$.

This set is open but I am not sure about the accumulation points.
 
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dwsmith said:
All points $(x,y)$ such that $x^2 - y^2 < 1$.

This set is open but I am not sure about the accumulation points.

The set is the region of \(\mathbb{R}^2\) between the branches of the hyperbola \(x^2-y^2=1\). All the points on the hyperbola are accumulation points, so the set of accumulation points is \( \{(x,y)\in \mathbb{R}^2: x^2-y^2\le 1 \}\)

CB
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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