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Example 2.30. Let T be the topology on N which contains Ø, N, and for each N E N with N > 0, the set {n € N | n > N}U{0}. Verify that Ts is a topology. Describe the open sets, closed sets, limit points of subsets, and sequential limit points of subsets of N with this topology.

Example 2.30. Let T be the topology on N which contains Ø, N, and for each N E N with N > 0, the set {n € N | n > N}U{0}. Verify that Ts is a topology. Describe the open sets, closed sets, limit points of subsets, and sequential limit points of subsets of N with this topology.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.2: Mappings
Problem 22E
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Please solve 2.30, 2.32 and 2.34 in detail please.

Definition 2.29. Let X be a space. Let A be a subset of X. We say that a point x E X is a sequential limit point of A iff there is a sequence {x;}1 in A\ {x} such that for every open set U in X containing x, there is an I EZ+ such that U contains x; for all i > I. Example 2.30. Let T be the topology on N which contains Ø, N, and for each N E N with N > 0, the set {n E N | n N}U{0}. Verify that Ts is a topology. Describe the open sets, closed sets, limit points of subsets, and sequential limit points of subsets of N with this topology. S Exercise 2.31. Let X be a space. Let A be a subset of X. Let x be a sequential limit point of A. Then x is a limit point of A. Problem 2.32. Is there an example of a space X, a subset Y CX, and a point x E X such that x is a limit point of Y, but not a sequential limit point of Y? 2.4. Subspace Topology. Sometimes we use a given topology on a set X to induce con- veniently a topology on a subset of X. Definition 2.33. Let X be a space with topology T and Y C X. T induces a topology TY on Y, called the subspace topology on Y, given by Ty := {V C Y |3U € T,V = UNY}. Problem 2.34. Let E C N be the set of even natural numbers. Give N the topology Tad (see Example 2.17). Describe the subspace topology on E.
Transcribed Image Text:Definition 2.29. Let X be a space. Let A be a subset of X. We say that a point x E X is a sequential limit point of A iff there is a sequence {x;}1 in A\ {x} such that for every open set U in X containing x, there is an I EZ+ such that U contains x; for all i > I. Example 2.30. Let T be the topology on N which contains Ø, N, and for each N E N with N > 0, the set {n E N | n N}U{0}. Verify that Ts is a topology. Describe the open sets, closed sets, limit points of subsets, and sequential limit points of subsets of N with this topology. S Exercise 2.31. Let X be a space. Let A be a subset of X. Let x be a sequential limit point of A. Then x is a limit point of A. Problem 2.32. Is there an example of a space X, a subset Y CX, and a point x E X such that x is a limit point of Y, but not a sequential limit point of Y? 2.4. Subspace Topology. Sometimes we use a given topology on a set X to induce con- veniently a topology on a subset of X. Definition 2.33. Let X be a space with topology T and Y C X. T induces a topology TY on Y, called the subspace topology on Y, given by Ty := {V C Y |3U € T,V = UNY}. Problem 2.34. Let E C N be the set of even natural numbers. Give N the topology Tad (see Example 2.17). Describe the subspace topology on E.
Example 2.17. On N, let {{0,1}, {2,3}, {4, 5}, . .} be a subset of a topology Tad: What other sets must be in Tdd at a minimum for it to be a topology on N?
Transcribed Image Text:Example 2.17. On N, let {{0,1}, {2,3}, {4, 5}, . .} be a subset of a topology Tad: What other sets must be in Tdd at a minimum for it to be a topology on N?
Expert Solution
Step 1

Definition: Let X be a non empty set and τ be a collection of subsets of X. Then X, τ is said to be a topological space if it satisfies the following conditions:

  • Empty set and the whole set in τ.
  • Arbitrary union of sets of τ is also in τ.
  • Intersection of finite (or two sets) sets of τ is also in τ.

Let τs be the topology on ℕ defined as follows

The empty set ∅, ℕ is in the topology. The set of the form n∈ℕ | n≥N∪0 is also in the topology for every N in ℕ.

If N=1, the set will be n∈ℕ | n≥1∪0, i.e.,

1, 2, 3, 4, …

If N=2, the set will be n∈ℕ | n≥2∪0, i.e.,

 2, 3, 4, …

If N=10, the set will be n∈ℕ | n≥10∪0, i.e.,

10, 11, 12, 13, …

Hence, if N is minimum the set will be larger.

Since τs contains the empty set and the whole set, it satisfies the first condition.

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