Linearly Ordered and Generalized Ordered Spaces
... though it is unlikely that their authors would have seen them as such. In 1941, Eilenberg published one of the first modern orderability theorems, proving that a connected, locally connected space X is orderable if and only if X 2 − {(x, x) : x ∈ X} is not connected. Kowalsky proved another orderab ...
... though it is unlikely that their authors would have seen them as such. In 1941, Eilenberg published one of the first modern orderability theorems, proving that a connected, locally connected space X is orderable if and only if X 2 − {(x, x) : x ∈ X} is not connected. Kowalsky proved another orderab ...
PDF
... You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 For connected manifolds, the assumption that M is second-countable is logically equivalent to M being paracompact, or equivalently to M being metrizable. The topological hyp ...
... You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 For connected manifolds, the assumption that M is second-countable is logically equivalent to M being paracompact, or equivalently to M being metrizable. The topological hyp ...
CONVERGENT SEQUENCES IN TOPOLOGICAL SPACES 1
... 5. On X = R consider the particular point topology Tp with p = 0. Then the only sequences converging to p = 0 are the sequences which are constant (and equal to zero) after some n0 , that is there is some n0 such that for all n ≥ n0 we have xn = 0. Describe all sequences which converge to x0 = 1! 2. ...
... 5. On X = R consider the particular point topology Tp with p = 0. Then the only sequences converging to p = 0 are the sequences which are constant (and equal to zero) after some n0 , that is there is some n0 such that for all n ≥ n0 we have xn = 0. Describe all sequences which converge to x0 = 1! 2. ...
Descriptive set theory, dichotomies and graphs
... Let E be a definable equivalence relation on a Hausdorff space (e.g. isomorphism relation on a suitable class of objects) then exactly one of following holds: There are at most countably many equivalence classes. There exists a perfect set of inequivalent elements. We will prove this from a graph th ...
... Let E be a definable equivalence relation on a Hausdorff space (e.g. isomorphism relation on a suitable class of objects) then exactly one of following holds: There are at most countably many equivalence classes. There exists a perfect set of inequivalent elements. We will prove this from a graph th ...
4.2 Notes
... Corollary: a corollary to a theorem is a statement that can be proved easily using the theorem. Corollary to the Triangle Sum Theorem: the acute angles of a right triangle are complimentary. In ABC , if mC 90, then mA mB 90 . B ...
... Corollary: a corollary to a theorem is a statement that can be proved easily using the theorem. Corollary to the Triangle Sum Theorem: the acute angles of a right triangle are complimentary. In ABC , if mC 90, then mA mB 90 . B ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.