Sung-Hoon Park - Quotient Topology
... Let p: X Y be a quotient map. Let Z be a space and let g: X Z be a map that is constant on each set p1(y), for y Y. then g induces a map f: Y Z such that f g=g. The induced map f is continuous if and only if g is continuous; f is a quotient map if and only if g is a quotient map. ...
... Let p: X Y be a quotient map. Let Z be a space and let g: X Z be a map that is constant on each set p1(y), for y Y. then g induces a map f: Y Z such that f g=g. The induced map f is continuous if and only if g is continuous; f is a quotient map if and only if g is a quotient map. ...
Order, topology, and preference
... quite well by its title. To the reader minimally knowledgeable in topology, this study In any case, Dugundji's ...
... quite well by its title. To the reader minimally knowledgeable in topology, this study In any case, Dugundji's ...
Axioms of separation - GMU Math 631 Spring 2011
... interval of ω1 is countable (otherwise we would have a contradiction with Proosition 34). So, if ω1 would have a countable cofinite subset, say A, then ω1 will be the union of countably many coutable subsets (initial intervals with the right endpoints at a ∈ A) and thus would be countable - a contra ...
... interval of ω1 is countable (otherwise we would have a contradiction with Proosition 34). So, if ω1 would have a countable cofinite subset, say A, then ω1 will be the union of countably many coutable subsets (initial intervals with the right endpoints at a ∈ A) and thus would be countable - a contra ...
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... of) the “reverse” path. Notice that for x ∈ X, the group of automorphisms of x is the fundamental group of X with basepoint x, HomΠ1 (X) (x, x) = π1 (X, x) . Definition 2. Let f : X → Y be a continuous function between two topological spaces. Then there is an induced functor Π1 (f ) : Π1 (X) → Π1 (Y ...
... of) the “reverse” path. Notice that for x ∈ X, the group of automorphisms of x is the fundamental group of X with basepoint x, HomΠ1 (X) (x, x) = π1 (X, x) . Definition 2. Let f : X → Y be a continuous function between two topological spaces. Then there is an induced functor Π1 (f ) : Π1 (X) → Π1 (Y ...
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 7 Contents
... real life, we’ll use (a variant of) the “naive” description. We begin by recasting morphisms of affines in a different light. Theorem. Let X ⊂ Am , Y ⊂ An be irreducible algebraic sets, and let π : X → Y be a continuous map. Then the following are equivalent. (Draw pictures!) i) π is a morphism of a ...
... real life, we’ll use (a variant of) the “naive” description. We begin by recasting morphisms of affines in a different light. Theorem. Let X ⊂ Am , Y ⊂ An be irreducible algebraic sets, and let π : X → Y be a continuous map. Then the following are equivalent. (Draw pictures!) i) π is a morphism of a ...
Some comments on Heisenberg-picture QFT, Theo Johnson
... Corollary: Let X be a scheme. (QCOH(X), OX ) ∈ are well-behaved for “tensoring”-type operations. In particular, for any commutative ring hom Z[q, q −1 ] → R, ALG0 (VECT2 ) is 1-dualizable iff X is (0-)affine. there is a category TLR⊗Z[q,q −1 ] R ∈ ALG2 (PRESR ), and The 1-dim TQFT defined by (QCOH(X ...
... Corollary: Let X be a scheme. (QCOH(X), OX ) ∈ are well-behaved for “tensoring”-type operations. In particular, for any commutative ring hom Z[q, q −1 ] → R, ALG0 (VECT2 ) is 1-dualizable iff X is (0-)affine. there is a category TLR⊗Z[q,q −1 ] R ∈ ALG2 (PRESR ), and The 1-dim TQFT defined by (QCOH(X ...
Primal spaces and quasihomeomorphisms - RiuNet
... Quasihomeomorphisms are used in algebraic geometry and it has recently been shown that this notion arises naturally in the theory of some foliations associated to closed connected manifolds (one may see [6]). A principal space is a topological space in which any intersection of open sets is open. It ...
... Quasihomeomorphisms are used in algebraic geometry and it has recently been shown that this notion arises naturally in the theory of some foliations associated to closed connected manifolds (one may see [6]). A principal space is a topological space in which any intersection of open sets is open. It ...
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... Let X be a set. A net is a map from a directed set to X. In other words, it is a pair (A, γ) where A is a directed set and γ is a map from A to X. If a ∈ A then γ(a) is normally written xa , and then the net is written (xa )a∈A , or simply (xa ) if the direct set A is understood. Now suppose X is a ...
... Let X be a set. A net is a map from a directed set to X. In other words, it is a pair (A, γ) where A is a directed set and γ is a map from A to X. If a ∈ A then γ(a) is normally written xa , and then the net is written (xa )a∈A , or simply (xa ) if the direct set A is understood. Now suppose X is a ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.