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... Sometimes we can use information about the product space X × X together with the diagonal embedding to get back information about X. For instance, X is Hausdorff if and only if the image of ∆ is closed in X × X [proof]. If we know more about the product space than we do about X, it might be easier t ...
... Sometimes we can use information about the product space X × X together with the diagonal embedding to get back information about X. For instance, X is Hausdorff if and only if the image of ∆ is closed in X × X [proof]. If we know more about the product space than we do about X, it might be easier t ...
Solutions for the Midterm Exam
... and so there are disjoint open sets U and V , containing x and y, respectively. By definition of the product topology, U × V is an open subset of X × X, and clearly U × V ⊂ ∆c (for otherwise U ∩ V 6= ∅). This shows that ∆c is open. Conversely, suppose ∆ is closed, that is to say, ∆c is open. Let x a ...
... and so there are disjoint open sets U and V , containing x and y, respectively. By definition of the product topology, U × V is an open subset of X × X, and clearly U × V ⊂ ∆c (for otherwise U ∩ V 6= ∅). This shows that ∆c is open. Conversely, suppose ∆ is closed, that is to say, ∆c is open. Let x a ...
Tutorial 12 - School of Mathematics and Statistics, University of Sydney
... α < β. Observe that the complement of this set is the closed and bounded interval [α, β]. An arbitrary open subset of R ∪ {∞} is a union of subsets corresponding to open arcs. If none of these arcs contain (0, 1) the result is simply an open subset of R, and we can obtain any open subset of R in thi ...
... α < β. Observe that the complement of this set is the closed and bounded interval [α, β]. An arbitrary open subset of R ∪ {∞} is a union of subsets corresponding to open arcs. If none of these arcs contain (0, 1) the result is simply an open subset of R, and we can obtain any open subset of R in thi ...
G-sets, G-spaces and Covering Spaces
... Example Given a subgroup H ⊂ G, denote by G/H the set of all cosets gH = {gh : h ∈ H} of H in G. It is a transitive G-set with action g 0 (gH) = (g 0 g)H. (This does not conflict with the notation of Definition 5: G/H is the orbit space of the action of H on G given by right multiplication, (h, g) 7 ...
... Example Given a subgroup H ⊂ G, denote by G/H the set of all cosets gH = {gh : h ∈ H} of H in G. It is a transitive G-set with action g 0 (gH) = (g 0 g)H. (This does not conflict with the notation of Definition 5: G/H is the orbit space of the action of H on G given by right multiplication, (h, g) 7 ...
