FINAL EXAM
... 1. (a) Suppose Y is a closed subspace of a topological space X, and A is a closed subspace of Y . Show that A is a closed subspace of X. (b) Suppose A is a closed subspace of X, and B is a closed subspace of Y . Show that A × B is a closed subspace of X × Y . 2. Let Y be a compact space, and X an ar ...
... 1. (a) Suppose Y is a closed subspace of a topological space X, and A is a closed subspace of Y . Show that A is a closed subspace of X. (b) Suppose A is a closed subspace of X, and B is a closed subspace of Y . Show that A × B is a closed subspace of X × Y . 2. Let Y be a compact space, and X an ar ...
The subspace topology, ctd. Closed sets and limit points.
... In R the singletons {x} are all closed sets. For the complement is (−∞, x) ∪ (x, ∞), which is open. In a general topological space X, the singletons need not be closed. For example if we endow R with the concrete topology, which has as its closed sets R and ∅, none of the singletons are closed. Anot ...
... In R the singletons {x} are all closed sets. For the complement is (−∞, x) ∪ (x, ∞), which is open. In a general topological space X, the singletons need not be closed. For example if we endow R with the concrete topology, which has as its closed sets R and ∅, none of the singletons are closed. Anot ...
9/21 handout
... (c) If µ is another topology on X and τ ≤ µ, then (xn ) → x =⇒ (xn ) → x. Proof : Homework, maybe. Examples: 1. In any discrete space, each singleton {x} is open, and in order for a sequence (xn ) to converge to x, (xn ) must be eventually in {x}, which means (xn ) must be eventually constant. 2. In ...
... (c) If µ is another topology on X and τ ≤ µ, then (xn ) → x =⇒ (xn ) → x. Proof : Homework, maybe. Examples: 1. In any discrete space, each singleton {x} is open, and in order for a sequence (xn ) to converge to x, (xn ) must be eventually in {x}, which means (xn ) must be eventually constant. 2. In ...
METRIC SPACES AND COMPACTNESS 1. Definition and examples
... To see completeness, let xi ∈ X be any Cauchy sequence in X. By theorem 5.5 there is a convergent subsequence of xi with limit x0 . It is then easy to see that xi converges with limit x0 (see exercise 3). Thus X is complete. On the other hand, consider the open cover Fε of X defined for any ε > 0 as ...
... To see completeness, let xi ∈ X be any Cauchy sequence in X. By theorem 5.5 there is a convergent subsequence of xi with limit x0 . It is then easy to see that xi converges with limit x0 (see exercise 3). Thus X is complete. On the other hand, consider the open cover Fε of X defined for any ε > 0 as ...
Notes - Ohio State Computer Science and Engineering
... Two topological spaces are considered to be the same if the points that comprise them are connected the same way. For example, the boundary of a cube can be deformed into a sphere without cutting or gluing it. They have the same topology. We formalize this notion of topological equality by defining ...
... Two topological spaces are considered to be the same if the points that comprise them are connected the same way. For example, the boundary of a cube can be deformed into a sphere without cutting or gluing it. They have the same topology. We formalize this notion of topological equality by defining ...
Problem set 1: Answers
... • For each pair of objects X, Y ∈ ob(C) and each morphism σ : X → Y in mor(X, Y ), there is a morphism F (σ) : F (X) → F (Y ) in mor(F (X), F (Y )). satisfying the following conditions: • Compatibility with identites. For each object X ∈ ob(C), F (idX ) = idF (X) • Compatibility with composition. F ...
... • For each pair of objects X, Y ∈ ob(C) and each morphism σ : X → Y in mor(X, Y ), there is a morphism F (σ) : F (X) → F (Y ) in mor(F (X), F (Y )). satisfying the following conditions: • Compatibility with identites. For each object X ∈ ob(C), F (idX ) = idF (X) • Compatibility with composition. F ...
PDF
... Then E is called a covering space, p is called a covering map, the Ei ’s are sheets of the covering of U and for each x ∈ X, p−1 (x) is the fiber of p above x. The open set U is said to be evenly covered. If E is simply connected, it is called the universal covering space. From this we can derive th ...
... Then E is called a covering space, p is called a covering map, the Ei ’s are sheets of the covering of U and for each x ∈ X, p−1 (x) is the fiber of p above x. The open set U is said to be evenly covered. If E is simply connected, it is called the universal covering space. From this we can derive th ...
1 Topological and metric spaces
... We proceed to show that completeness together with totally boundedness imply compactness. Let U be a complete and totally bounded subset. Assume U is not compact and choose a covering {Uα }α∈A of U that does not admit a nite subcovering. On the other hand, U is totally bounded and admits a covering ...
... We proceed to show that completeness together with totally boundedness imply compactness. Let U be a complete and totally bounded subset. Assume U is not compact and choose a covering {Uα }α∈A of U that does not admit a nite subcovering. On the other hand, U is totally bounded and admits a covering ...
1.8 Completeness - Matrix Editions
... Proof. 1 =⇒ 2 : Note that statement 1 implies Theorem 1.7.30. Let (xn ) be a Cauchy sequence in R. Then (xn ) is bounded by part 1 of Proposition 1.7.17. By Theorem 1.7.29, (xn ) contains a monotone subsequence (xnk ). Since (xnk ) is a bounded monotone sequence in R, it must converge by Theorem 1.7 ...
... Proof. 1 =⇒ 2 : Note that statement 1 implies Theorem 1.7.30. Let (xn ) be a Cauchy sequence in R. Then (xn ) is bounded by part 1 of Proposition 1.7.17. By Theorem 1.7.29, (xn ) contains a monotone subsequence (xnk ). Since (xnk ) is a bounded monotone sequence in R, it must converge by Theorem 1.7 ...
Math 396. Gluing topologies, the Hausdorff condition, and examples
... {Sα } is a collection of subsets of X in τ then we need the union S of the Sα ’s to be in τ . That is, we want S ∩ Xi to be τi -open for each i. This overlap is the union of the overlaps Sα ∩ Xi that are τi -open for all α (due to the assumption that every Sα is in τ ), and so this union is also τi ...
... {Sα } is a collection of subsets of X in τ then we need the union S of the Sα ’s to be in τ . That is, we want S ∩ Xi to be τi -open for each i. This overlap is the union of the overlaps Sα ∩ Xi that are τi -open for all α (due to the assumption that every Sα is in τ ), and so this union is also τi ...
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.