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... points in that manifold Qm , the configuration space Fm,n M is the space of ordered n-tuples {x1 , ..., xn } ∈ M − Qm such that for all i, j ∈ {1, ..., n}, with i 6= j, xi 6= xj . If M is simply connected, then the choice of the m points is irrelevant, as, for any two such sets Qm and Q0m , one has ...
... points in that manifold Qm , the configuration space Fm,n M is the space of ordered n-tuples {x1 , ..., xn } ∈ M − Qm such that for all i, j ∈ {1, ..., n}, with i 6= j, xi 6= xj . If M is simply connected, then the choice of the m points is irrelevant, as, for any two such sets Qm and Q0m , one has ...
Let X be a metric space and R the additive group of the reals
... Proof. Let x ∈ X. Suppose that γ(x) is not a one-to-one image of R, i.e., there are s0 and s00 in R, with s0 > s00 such that ϕ(s0, x) = ϕ(s00, x). Then ϕ(t, x) = x, where t = s0 − s00. Let t0 = inf{t > 0 | ϕ(t, x) = x}. Then either ϕ(t0, x) = x or there is a sequence (tn)tn>t0 convergent to t0 such ...
... Proof. Let x ∈ X. Suppose that γ(x) is not a one-to-one image of R, i.e., there are s0 and s00 in R, with s0 > s00 such that ϕ(s0, x) = ϕ(s00, x). Then ϕ(t, x) = x, where t = s0 − s00. Let t0 = inf{t > 0 | ϕ(t, x) = x}. Then either ϕ(t0, x) = x or there is a sequence (tn)tn>t0 convergent to t0 such ...
Note on the Tychonoff theorem and the axiom of choice.
... terms of roominess. For example, if you squeeze an infinite set of points into the unit interval, they get cramped—for any $ > 0, there are two points that are less than $ apart. But, it’s easy to fit an infinite number of points in the real line so that they’re all spread out. This idea is summariz ...
... terms of roominess. For example, if you squeeze an infinite set of points into the unit interval, they get cramped—for any $ > 0, there are two points that are less than $ apart. But, it’s easy to fit an infinite number of points in the real line so that they’re all spread out. This idea is summariz ...
bornological countable enlargements
... Remark 2.8. This corollary shows in particular that Theorem 2.4 and Corollary 2.5 are no longer valid in general if CE is replaced by arbitrary enlargement. 3. Bornological and quasibarrelled CEs Our two main results on bornological spaces arise as corollaries to the following general theorem. Recal ...
... Remark 2.8. This corollary shows in particular that Theorem 2.4 and Corollary 2.5 are no longer valid in general if CE is replaced by arbitrary enlargement. 3. Bornological and quasibarrelled CEs Our two main results on bornological spaces arise as corollaries to the following general theorem. Recal ...
α OPEN SETS IN TRI TOPOLOGICAL SPACE
... In 1961 J .C. Kelly [ II] introduced the concept of bitopolgical space. N. Levine [VI] introduced the idea of semi open sets and semi continuity and Mashhour et. al [ VII ] introduced the concept of pre open sets and pre continuity in a topological space. F.H. Khedr , S.M. Al-Areefi and T. Noiri [IV ...
... In 1961 J .C. Kelly [ II] introduced the concept of bitopolgical space. N. Levine [VI] introduced the idea of semi open sets and semi continuity and Mashhour et. al [ VII ] introduced the concept of pre open sets and pre continuity in a topological space. F.H. Khedr , S.M. Al-Areefi and T. Noiri [IV ...
First-Order Logical Duality Henrik Forssell
... Then CA can be recovered from the category ModA of models as the category / Sets that preserve all HomG (ModA , Sets) of those functors F : ModA limits, filtered colimits, and regular epimorphisms, CA ' HomG (ModA , Sets) We think of those as the ‘continuous’ maps in this context. In a wider perspe ...
... Then CA can be recovered from the category ModA of models as the category / Sets that preserve all HomG (ModA , Sets) of those functors F : ModA limits, filtered colimits, and regular epimorphisms, CA ' HomG (ModA , Sets) We think of those as the ‘continuous’ maps in this context. In a wider perspe ...