Topology III Exercise set 6 1. Show that the following are equivalent
... 1. Show that the following are equivalent for a topological space X: A. X is normal. B. Every finite open cover of X has a finite open point-star refinement. C. Every locally finite open cover of X has a locally finite open point-star refinement. 2. (a) Show that, for every open cover U of the ordin ...
... 1. Show that the following are equivalent for a topological space X: A. X is normal. B. Every finite open cover of X has a finite open point-star refinement. C. Every locally finite open cover of X has a locally finite open point-star refinement. 2. (a) Show that, for every open cover U of the ordin ...
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... 1. Recall that the product topolgy is defined on the topologies X and Y as follows: A subset U ⊆ X × Y is open in X × Y if for all (x, y) ∈ U , there exists UX (open in X) and UY (open in Y ) so that (x, y) ∈ UX × UY ⊆ U . Please show that the standard topology on R2 is the same topology as the prod ...
... 1. Recall that the product topolgy is defined on the topologies X and Y as follows: A subset U ⊆ X × Y is open in X × Y if for all (x, y) ∈ U , there exists UX (open in X) and UY (open in Y ) so that (x, y) ∈ UX × UY ⊆ U . Please show that the standard topology on R2 is the same topology as the prod ...
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... to f (x). The concept of net can be replaced by the more familiar one of sequence if the spaces X and Y are first countable. 6. Whenever two nets S and T in X converge to the same point, then f ◦ S and f ◦ T converge to the same point in Y . 7. If F is a filter on X that converges to x, then f (F) i ...
... to f (x). The concept of net can be replaced by the more familiar one of sequence if the spaces X and Y are first countable. 6. Whenever two nets S and T in X converge to the same point, then f ◦ S and f ◦ T converge to the same point in Y . 7. If F is a filter on X that converges to x, then f (F) i ...
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.