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Why are the sets U and V pictured open? My understanding is that X is inheriting the subspace topology from R^2. So the basis elements are rectangles of R^2 intersecting with the
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real analysis - Set S=$\left\{ \left(x,y\right);x^{2}+y^{2}\leq1,x<1\right\} $ is open or closed? - Mathematics Stack Exchange
Topological space. Topology. Open and closed sets. Neighborhood. Interior, exterior, limit, boundary, isolated point. Dense, nowhere dense set.