D-FORCED SPACES: A NEW APPROACH TO RESOLVABILITY 1
... in ZFC. Our theorems 4.5 and 4.8 give a large number of 0-dimensional T2 (and so Tychonov) counterexamples in ZFC. The question if this can be done has been asked much more recently again in [8] and [10]. Our results are obtained with the help of a new method that is presented in section 2. Here we ...
... in ZFC. Our theorems 4.5 and 4.8 give a large number of 0-dimensional T2 (and so Tychonov) counterexamples in ZFC. The question if this can be done has been asked much more recently again in [8] and [10]. Our results are obtained with the help of a new method that is presented in section 2. Here we ...
1. Theorem: If (X,d) is a metric space, then the following are
... Then for each n ∈ Z + define Un = {Bd (x, 1/n)|x ∈ X}. Now, clearly each Un covers X, so we can find a countable subcollection of each that covers X. Also, since n is fixed for each Un , the countability of the subcollection must come from a countable subset, Dn ⊆ X, over which we now index. Thus th ...
... Then for each n ∈ Z + define Un = {Bd (x, 1/n)|x ∈ X}. Now, clearly each Un covers X, so we can find a countable subcollection of each that covers X. Also, since n is fixed for each Un , the countability of the subcollection must come from a countable subset, Dn ⊆ X, over which we now index. Thus th ...
Strong transitivity properties for operators arXiv
... Definition 1.1. We say that a non-empty collection F of subsets of Z+ is a family provided that each set A ∈ F is infinite and that F is hereditarily upward (i.e. for any A ∈ F , if B ⊃ A then B ∈ F ). The dual family F ∗ of F is defined as the collection of subsets A of Z+ such that A ∩ B 6= ∅ for ...
... Definition 1.1. We say that a non-empty collection F of subsets of Z+ is a family provided that each set A ∈ F is infinite and that F is hereditarily upward (i.e. for any A ∈ F , if B ⊃ A then B ∈ F ). The dual family F ∗ of F is defined as the collection of subsets A of Z+ such that A ∩ B 6= ∅ for ...
Unitary Group Actions and Hilbertian Polish
... be the group of all unitary tranformations of H. When U (H) is endowed with the strong operator topology it becomes a Polish group, and we denote it by U ∞ (following [7]). A more standard notation for this topological group is U (H)s . However, the notation U∞ stresses our interest in the abstract ...
... be the group of all unitary tranformations of H. When U (H) is endowed with the strong operator topology it becomes a Polish group, and we denote it by U ∞ (following [7]). A more standard notation for this topological group is U (H)s . However, the notation U∞ stresses our interest in the abstract ...
Stability of convex sets and applications
... set of quantum states in a separable Hilbert space [11]. A more general example is the convex set of all Borel probability measures on an arbitrary complete separable metric space endowed with the topology of weak convergence (the µ-compactness and stability of this set are proved in [10], Corollary ...
... set of quantum states in a separable Hilbert space [11]. A more general example is the convex set of all Borel probability measures on an arbitrary complete separable metric space endowed with the topology of weak convergence (the µ-compactness and stability of this set are proved in [10], Corollary ...
Algebraic Geometry, autumn term 2015
... of a certain type in mathematics, which are often sets with an additional structure, one should at the same time study “maps” between those objects that preserve the given structure. Thus one studies vector spaces along with linear maps, groups along with group homomorphisms, rings with ring homomor ...
... of a certain type in mathematics, which are often sets with an additional structure, one should at the same time study “maps” between those objects that preserve the given structure. Thus one studies vector spaces along with linear maps, groups along with group homomorphisms, rings with ring homomor ...
Compact operators on Banach spaces
... For infinite-dimensional Banach spaces, 0 inevitably lies in the spectrum, otherwise T would be invertible. Then 1 = T ◦ T −1 is the composition of a compact operator and a continuous operator, so is continuous, possible only in finite-dimensional spaces. Suppose there were infinitely-many different ...
... For infinite-dimensional Banach spaces, 0 inevitably lies in the spectrum, otherwise T would be invertible. Then 1 = T ◦ T −1 is the composition of a compact operator and a continuous operator, so is continuous, possible only in finite-dimensional spaces. Suppose there were infinitely-many different ...
General Topology
... in the form A ∋ x. So, the origin of notation is sort of ignored, but a more meaningful similarity to the inequality symbols < and > is emphasized. To state that x is not an element of A, we write x 6∈ A or A 6∋ x. § 1 ◦ 2 Equality of Sets A set is determined by its elements. It is nothing but a col ...
... in the form A ∋ x. So, the origin of notation is sort of ignored, but a more meaningful similarity to the inequality symbols < and > is emphasized. To state that x is not an element of A, we write x 6∈ A or A 6∋ x. § 1 ◦ 2 Equality of Sets A set is determined by its elements. It is nothing but a col ...
