Exercise Sheet no. 1 of “Topology”
... Let p be a prime number. For q ∈ Q× we define |q|p := p−n if we can write q = pn ab , where a ∈ Z, 0 6= b ∈ Z are not multiples of p. Note that this determines a unique n ∈ Z. We also put |0|p := 0. Show that d(x, y) := |x − y|p defines a metric on Q for which the sequence (pn )n∈N converges to 0. E ...
... Let p be a prime number. For q ∈ Q× we define |q|p := p−n if we can write q = pn ab , where a ∈ Z, 0 6= b ∈ Z are not multiples of p. Note that this determines a unique n ∈ Z. We also put |0|p := 0. Show that d(x, y) := |x − y|p defines a metric on Q for which the sequence (pn )n∈N converges to 0. E ...
Math 446–646 Important facts about Topological Spaces
... (c) Hausdorff property or property T2 : A topological space (X, T ) has the Hausdorff property if given two different point x, y ∈ X, there exist open sets O1 , O2 ∈ T such that x ∈ O1 , y ∈ O2 and O1 ∩ O2 = ∅. Any space with the property T2 also has property T1 , but the converse is not true. If (X ...
... (c) Hausdorff property or property T2 : A topological space (X, T ) has the Hausdorff property if given two different point x, y ∈ X, there exist open sets O1 , O2 ∈ T such that x ∈ O1 , y ∈ O2 and O1 ∩ O2 = ∅. Any space with the property T2 also has property T1 , but the converse is not true. If (X ...
Problem 1: We denote the usual “Euclidean” metric on IRn by de : |x
... (a) Let X be a set and Tα , α ∈ A, be a family of topologies on X. Show that α∈A Tα is a topology on X. (b) Let S be a family of subsets of X. Show that there is a topology T on X such that S ⊂ T and if T̂ is another topology on X with the the property S ⊂ T̂ , then T ⊂ T̂ . That is, T is the smalle ...
... (a) Let X be a set and Tα , α ∈ A, be a family of topologies on X. Show that α∈A Tα is a topology on X. (b) Let S be a family of subsets of X. Show that there is a topology T on X such that S ⊂ T and if T̂ is another topology on X with the the property S ⊂ T̂ , then T ⊂ T̂ . That is, T is the smalle ...
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... 3. DISprove the following assertion: Let τ and τ 0 be two topologies on the same space X. Then if (X, τ ) ≈ (X, τ 0 ), then τ = τ 0 . 4. A discrete map is a continuous function d : X → D, where D is a finite set given the discrete topology. Prove that X is connected if and only if every discrete map ...
... 3. DISprove the following assertion: Let τ and τ 0 be two topologies on the same space X. Then if (X, τ ) ≈ (X, τ 0 ), then τ = τ 0 . 4. A discrete map is a continuous function d : X → D, where D is a finite set given the discrete topology. Prove that X is connected if and only if every discrete map ...
SEPARATION AXIOMS 1. The axioms The following categorization
... (3) Consider X = R with the excluded point topology T p with p = 0. This space is T0 but fails to be T1 since the only open set containing p is X. (4) Let X = R be given the lower limit topology Tll . Then X is Ti for every i = 0, 1, ..., 5. To see this is suffices to show that it is T2 and T5 (all ...
... (3) Consider X = R with the excluded point topology T p with p = 0. This space is T0 but fails to be T1 since the only open set containing p is X. (4) Let X = R be given the lower limit topology Tll . Then X is Ti for every i = 0, 1, ..., 5. To see this is suffices to show that it is T2 and T5 (all ...
Topological Spaces
... The topology T = 2X , which is generated by the discrete metric, is the finest of all. This topology is called the discrete topology. At the other extreme, the topology consisting of φ and X, which is called the indiscrete topology is the coarsest of all. Between these extremes, the various topologi ...
... The topology T = 2X , which is generated by the discrete metric, is the finest of all. This topology is called the discrete topology. At the other extreme, the topology consisting of φ and X, which is called the indiscrete topology is the coarsest of all. Between these extremes, the various topologi ...
Problem Set 1 - Columbia Math
... 1. Show that collection τ0 of subsets of Rn forms a topology on Rn . 2. Let S = [0, 1] ⊂ R. Equip S with the subspace topology as a subset of R. Show that the set (1/2, 1] is open in S but that (1/2, 1] is not open in R. 3. This exercise will show that the product topology on R2 coincides with the s ...
... 1. Show that collection τ0 of subsets of Rn forms a topology on Rn . 2. Let S = [0, 1] ⊂ R. Equip S with the subspace topology as a subset of R. Show that the set (1/2, 1] is open in S but that (1/2, 1] is not open in R. 3. This exercise will show that the product topology on R2 coincides with the s ...
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.