Functional Analysis
... Problem 1 (Hausdorff spaces). a) Let (X, T ) be a Hausdorff space and Y ⊆ X. Prove that (Y, TY ) is a Hausdorff space, where TY is the relative topology of Y wrt. X. b) Let X := {(x, y) ∈ R2 | x2 + y 2 ≤ 1} and Cr := {(x, y) ∈ X | r < x2 + y 2 ≤ 1} for r ∈ [0, 1]. Prove that B := {Cr | r ∈ [0, 1]} ∪ ...
... Problem 1 (Hausdorff spaces). a) Let (X, T ) be a Hausdorff space and Y ⊆ X. Prove that (Y, TY ) is a Hausdorff space, where TY is the relative topology of Y wrt. X. b) Let X := {(x, y) ∈ R2 | x2 + y 2 ≤ 1} and Cr := {(x, y) ∈ X | r < x2 + y 2 ≤ 1} for r ∈ [0, 1]. Prove that B := {Cr | r ∈ [0, 1]} ∪ ...
Math 131: Midterm Solutions
... (1) Let X be a completely regular topological space. Let A and B be closed subsets of X with A ∩ B = ∅, and suppose that A is compact. Show that there exists a continuous function f : X → [0, 1] such that f (a) = 0 for a ∈ A and f (b) = 1 for b ∈ B. Since X is completely regular, we can choose for e ...
... (1) Let X be a completely regular topological space. Let A and B be closed subsets of X with A ∩ B = ∅, and suppose that A is compact. Show that there exists a continuous function f : X → [0, 1] such that f (a) = 0 for a ∈ A and f (b) = 1 for b ∈ B. Since X is completely regular, we can choose for e ...
(JJMS) 5(3), 2012, pp.201 - 208 g
... (2) For each closed set A and for each open set V containing A, there exists an Ig∗α -open set U such that A ⊆ U ⊆ cl∗α (U ) ⊆ V. Proof. (1) ⇒ (2) : Let A be a closed subset of X and B be an open set such that A ⊆ B. Since A and X −B are disjoint closed sets in X, there exists disjoint Ig∗α -open se ...
... (2) For each closed set A and for each open set V containing A, there exists an Ig∗α -open set U such that A ⊆ U ⊆ cl∗α (U ) ⊆ V. Proof. (1) ⇒ (2) : Let A be a closed subset of X and B be an open set such that A ⊆ B. Since A and X −B are disjoint closed sets in X, there exists disjoint Ig∗α -open se ...
Here
... • Topology is the study of qualitative/global aspects of shapes, or – more generally – the study of qualitative/global aspects in mathematics. A simple example of a ‘shape’ is a 2-dimensional surface in 3-space, like the surface of a ball, a football, or a donut. While a football is different from a ...
... • Topology is the study of qualitative/global aspects of shapes, or – more generally – the study of qualitative/global aspects in mathematics. A simple example of a ‘shape’ is a 2-dimensional surface in 3-space, like the surface of a ball, a football, or a donut. While a football is different from a ...
KUD Organizer
... That a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations. (Two-dimensional figures are similar if the second can be obtained by these methods and dilations.) Geometric figures can be transformed, analy ...
... That a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations. (Two-dimensional figures are similar if the second can be obtained by these methods and dilations.) Geometric figures can be transformed, analy ...
Multiscale virus capsid modeling
... Viruses are contagious agents and can cause epidemics and pandemics They grow and/or reproduce inside a host cell. Figure 1 shows four typical virus morphologies. Virus infection starts with the attachment of a virus on the host cell surface, with possible fusion of viral capsid surface and the host ...
... Viruses are contagious agents and can cause epidemics and pandemics They grow and/or reproduce inside a host cell. Figure 1 shows four typical virus morphologies. Virus infection starts with the attachment of a virus on the host cell surface, with possible fusion of viral capsid surface and the host ...
Part1 - Faculty
... Let (M ; T) be a topological space. A subset B of the topology T is called a basis of the topology if T contains exactly those sets which result from arbitrary unions of elements of B. Generating set of a Topology A subset E of the power set P(M) is called a generating set on the underlying set ...
... Let (M ; T) be a topological space. A subset B of the topology T is called a basis of the topology if T contains exactly those sets which result from arbitrary unions of elements of B. Generating set of a Topology A subset E of the power set P(M) is called a generating set on the underlying set ...
Document
... ←The proof in the other direction is analogous. Suppose the intersection of any centered system of closed subsets of X is nonempty. To prove that X is compact, let {Fi: i I} be a collection of open sets in X that cover X . We claim that this collection contains a finite subcollection that also co ...
... ←The proof in the other direction is analogous. Suppose the intersection of any centered system of closed subsets of X is nonempty. To prove that X is compact, let {Fi: i I} be a collection of open sets in X that cover X . We claim that this collection contains a finite subcollection that also co ...