MT 3810 3803
... Show that every compact metric space has the Bolzano-Weirstrass property. (OR) State and prove Tychanoff's Theorem. ...
... Show that every compact metric space has the Bolzano-Weirstrass property. (OR) State and prove Tychanoff's Theorem. ...
Algebraic Topology Introduction
... Every open set in Ā has the form f −1 (B) for some open B in Y . Then, apply f to each open set in Ā, f ◦ f −1 (B) = B since f surjects onto f (X). This finite collection of B’s forms a finite open cover of f (X), and therefore f (X) is compact. (4) Since Y is Hausdorff, a compact subset C of Y m ...
... Every open set in Ā has the form f −1 (B) for some open B in Y . Then, apply f to each open set in Ā, f ◦ f −1 (B) = B since f surjects onto f (X). This finite collection of B’s forms a finite open cover of f (X), and therefore f (X) is compact. (4) Since Y is Hausdorff, a compact subset C of Y m ...
PDF
... • every principal ideal is generated by an idempotent. When R = C(X), then R is commutative reduced with 1. In addition to the algebraic characterizations of R above, X being a P-space is equivalent to any of the following statements: • every zero set is open • if f, g ∈ C(X), then (f, g) = (f 2 + g ...
... • every principal ideal is generated by an idempotent. When R = C(X), then R is commutative reduced with 1. In addition to the algebraic characterizations of R above, X being a P-space is equivalent to any of the following statements: • every zero set is open • if f, g ∈ C(X), then (f, g) = (f 2 + g ...
Solutions to exercises in Munkres
... (4) Closed subsets of Y always have closed saturations as we saw in item (1). If f is closed then also the saturation, B ∪ f −1 f (A ∩ B) ∪ f (A ∩ B) ⊂ X ∪ Y , of a closed subset B ⊂ X is closed. (Since closed quotient maps (surjective closed maps) preserve normality [Ex 31.6, Thm 73.3] this gives a ...
... (4) Closed subsets of Y always have closed saturations as we saw in item (1). If f is closed then also the saturation, B ∪ f −1 f (A ∩ B) ∪ f (A ∩ B) ⊂ X ∪ Y , of a closed subset B ⊂ X is closed. (Since closed quotient maps (surjective closed maps) preserve normality [Ex 31.6, Thm 73.3] this gives a ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.