/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Prove that in a discrete topolog... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 6

Prove that in a discrete topological space, each subset is simultaneously open and closed.

Short Answer

Expert verified
In a discrete space, every subset is both open and closed because all subsets, including complements, are in the topology.

Step by step solution

01

Understand the Definitions

Recall the definitions of a discrete topological space and the properties of open and closed sets. In a discrete topological space, every subset is part of the topology, meaning it is open. A set is closed if its complement is open.
02

Show Subsets are Open

By definition, in a discrete topological space, the topology is comprised of all possible subsets. Therefore, no matter which subset you select, it is included in the topology and thus is open.
03

Demonstrate Subsets are Closed

To show a set is closed, its complement must be open. In a discrete space, since every subset (and thus every complement) is also in the topology, the complement of any subset is open.
04

Conclude subsets are both open and closed

Since every subset in a discrete topological space is open due to being a part of the topology, and every subset is closed because its complement is also in the topology and therefore open, each subset is both open and closed.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Open Sets
In topology, an open set is a fundamental concept that helps us understand the structure of a space.
To determine if a set is open, we need to consider the broader topology in which the set resides.
In the context of a discrete topological space, the situation simplifies significantly.
  • In a discrete topological space, all possible subsets of the space are considered open.
  • This means that regardless of which subset you choose, it will always be classified as an open set in this space.
The intuition behind this is that in a discrete topology, each point stands alone in its own little 'bubble' without any restrictions imposed by neighboring points.
Therefore, every collection of points, no matter how small or large, is open.
This makes discrete spaces particularly simple to analyze compared to more complex topological structures.
Closed Sets
Closed sets are central to discussions about continuity and convergence in topology.
A set is defined as closed if its complement is open within the given topology.
This definition ties directly to the idea of open sets.
  • In discrete topological spaces, as every subset is open, the complement of any subset is also open.
  • Consequently, this property ensures that every subset can be considered closed as well.
This duality arises from the simplicity of the discrete topology where no subset is excluded from being part of the topology.
Unlike in more intricate topologies, there is no distinction between being open and closed in discrete spaces.
Such properties make discrete topological spaces a convenient and familiar starting point for those new to topology.
Topology of Subsets
The topology of subsets refers to the collection of all open sets within a topological space.
In the case of discrete topological spaces, this collection is straightforward.
  • Every possible subset of the space is included in the topology, making it comprehensive and easy to work with.
  • This means that the topology on any subset of a discrete space mirrors the entire discrete topology.
Such a full, inclusive topology allows us to see each subset as a topological space in its own right, maintaining the properties of discrete topology.
This means every subset adheres to the same rules regarding openness and closedness as the overall space.
For students learning about topological spaces, discrete spaces provide a clear and understandable framework due to their straightforward nature.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \((X, \jmath)\) be a topological space that is metrizable. Prove that each neighborhood \(N\) of a point \(a \in X\) contains a neighborhood \(V\) of \(a\) such that \(V\) is a closed set.

. In the real line, prove that the boundary of the open interval \((a, b)\) and the boundary of the closed interval \([a, b]\) is \(\\{a, b\\}\).

Prove that the family of open intervals with rational end points is a basis for the topology of the real line.

Let \(C_{s}\) be the category of sets and functions. Verify that the set of equivalences in \(H(X, X)\) with the same rule of composition as in \(C_{s}\) is the group of one-one mappings of \(X\) onto itself. In general, verify that in any category \(C\) for each object \(X\), the set of equivalences in \(H(X, X)\) with the same rule of composition is a group.

Let \(Y\) be a subspace of \(X\) and let \(A\) be a subset of \(Y\). Denote by Int \(_{X}(A)\) the interior of \(A\) in the topological space \(X\) and by \(\operatorname{Int}_{Y}(A)\) the interior of \(A\) in the topological space \(Y\). Prove that \(\operatorname{Int}_{X}(A) \subset \operatorname{Int}_{Y}(A)\). Illustrate by an example the fact that in general \(\operatorname{Int}_{X}(A) \neq \operatorname{Int}_{Y}(A)\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.