3. Geometric Notions
... general topological space, convergence of a sequence can be very strange. For example, consider the following topology on a nonempty set X: Let x0 ∈ X be chosen once and for all. Define TIP = {∅ or U ⊂ X with x0 ∈ U }. This set of subsets determines a topology on X called the included point topology ...
... general topological space, convergence of a sequence can be very strange. For example, consider the following topology on a nonempty set X: Let x0 ∈ X be chosen once and for all. Define TIP = {∅ or U ⊂ X with x0 ∈ U }. This set of subsets determines a topology on X called the included point topology ...
II.1 Separation Axioms
... is an embedding. 명제 2 X is a Hausdorff space if and only if the diagonal ∆ = {(x, x) | x ∈ X} is closed in X × X. 증명 X is Hausdorff. ⇔ ∀(x, y) ∈ 4c , ∃ Open neighborhoods Ux , Uy of x and y s.t. Ux × Uy ⊂ 4c . ⇔ 4c is open in X × X. ⇔ 4 is closed in X × X. 명제 3 Suppose that X is Hausdorff, then the ...
... is an embedding. 명제 2 X is a Hausdorff space if and only if the diagonal ∆ = {(x, x) | x ∈ X} is closed in X × X. 증명 X is Hausdorff. ⇔ ∀(x, y) ∈ 4c , ∃ Open neighborhoods Ux , Uy of x and y s.t. Ux × Uy ⊂ 4c . ⇔ 4c is open in X × X. ⇔ 4 is closed in X × X. 명제 3 Suppose that X is Hausdorff, then the ...
MA5209L4 - Maths, NUS - National University of Singapore
... A category C consists of the following three mathematical entities: •A class ob(C), whose elements are called objects; •A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morph ...
... A category C consists of the following three mathematical entities: •A class ob(C), whose elements are called objects; •A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morph ...
Catalogue of Useful Topological Vectorspaces
... The space D of test functions on Rn is the collection of all compactly-supported smooth (i.e., infinitelydifferentiable) functions on Rn . It is the direct limit (ascending union) of the Frechet spaces DK of smooth functions with support in a given compact set K. It is locally convex, but not metriz ...
... The space D of test functions on Rn is the collection of all compactly-supported smooth (i.e., infinitelydifferentiable) functions on Rn . It is the direct limit (ascending union) of the Frechet spaces DK of smooth functions with support in a given compact set K. It is locally convex, but not metriz ...
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.