Mathematics Exam > Mathematics Notes > Mathematics for IIT JAM, GATE, CSIR NET, UGC NET > Subspace Topology - Topology, CSIR-NET Mathematical Sciences
Subspace Topology - Topology, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download
We now consider some ways of getting new topologies from old ones.
Definition
If A is a subset of a topological space (X, JX), we define the subspace topology JA on A by:
B ∈ JA if B = A ∩ C for some C ∈ JX .
Examples
Restricting the metric on a metric space to a subset gives this topology.
For example, On X = [0, 1] with the usual metric inherited from R, the open sets are the intersection of [0, 1] with open sets of R.
So, for instance, [1, 1/4) = (-1, 1/4) ∩ [0, 1] and so is an open subset of the subspace X.
Remark
Note that as in this example, sets which are open in the subspace are not necessarily open in the "big space".- The subspace topology on Z ⊂ R (with its usual topology/metric) is the discrete topology.
- The subspace topology on the x-axis as a subset of R2 (with its usual topology) is the usual topology on R.
Remark
If we take the inclusion map i: A → X then the subspace topology is the weakest topology (fewest open sets) on A in which this map is continuous.
Proof
If B ⊂ X is open then i-1(B) = A ∩ B.
The document Subspace Topology - Topology, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is a part of the Mathematics Course Mathematics for IIT JAM, GATE, CSIR NET, UGC NET.
All you need of Mathematics at this link: Mathematics
|
556 videos|198 docs
|
FAQs on Subspace Topology - Topology, CSIR-NET Mathematical Sciences - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the subspace topology? |  |
Ans. The subspace topology is a topology that is induced on a subset of a topological space. It is a way to define a topology on a subset by considering only the open sets of the larger space that intersect with the subset.
| 2. How is the subspace topology defined? | |
Ans. The subspace topology on a subset A of a topological space X is defined by taking the collection of all sets of the form A ∩ U, where U is an open set in X. These sets form a basis for the subspace topology on A.
| 3. What are the properties of the subspace topology? | |
Ans. The subspace topology inherits certain properties from the larger topological space. For example, if X is Hausdorff, then any subspace of X is also Hausdorff. Similarly, if X is compact, then any subspace of X is also compact. However, the subspace topology may not be connected or path-connected even if the larger space is.
| 4. How is the subspace topology related to continuity? | |
Ans. A function between two topological spaces is continuous if and only if the inverse image of every open set in the codomain is open in the domain. In the case of a subspace, a function is continuous with respect to the subspace topology if and only if its restriction to the subspace is continuous in the larger space.
| 5. Can a subspace have a different topology than the larger space? | |
Ans. Yes, a subspace can have a different topology than the larger space. The subspace topology is a way to define a topology on a subset, and it is not necessarily the same as the topology of the larger space. The subspace topology only considers the open sets of the larger space that intersect with the subset, so it may have fewer open sets than the larger space.
Related Exams
About this Document
|
4.72/5 Rating |
|
Nov 24, 2024 Last updated |
Document Description: Subspace Topology - Topology, CSIR-NET Mathematical Sciences for Mathematics 2024 is part of Mathematics for IIT JAM, GATE, CSIR NET, UGC NET preparation.
The notes and questions for Subspace Topology - Topology, CSIR-NET Mathematical Sciences have been prepared according to the Mathematics exam syllabus. Information about Subspace Topology - Topology, CSIR-NET Mathematical Sciences covers topics
like and Subspace Topology - Topology, CSIR-NET Mathematical Sciences Example, for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Subspace Topology - Topology, CSIR-NET Mathematical Sciences.
Introduction of Subspace Topology - Topology, CSIR-NET Mathematical Sciences in English is available as part of our Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
for Mathematics & Subspace Topology - Topology, CSIR-NET Mathematical Sciences in Hindi for Mathematics for IIT JAM, GATE, CSIR NET, UGC NET course.
Download more important topics related with notes, lectures and mock test series for Mathematics
Exam by signing up for free. Mathematics: Subspace Topology - Topology, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
Description
Full syllabus notes, lecture & questions for Subspace Topology - Topology, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET - Mathematics | Plus excerises question with solution to help you revise complete syllabus for Mathematics for IIT JAM, GATE, CSIR NET, UGC NET | Best notes, free PDF download
Information about Subspace Topology - Topology, CSIR-NET Mathematical Sciences
In this doc you can find the meaning of Subspace Topology - Topology, CSIR-NET Mathematical Sciences defined & explained in the simplest way possible. Besides explaining types of
Subspace Topology - Topology, CSIR-NET Mathematical Sciences theory, EduRev gives you an ample number of questions to practice Subspace Topology - Topology, CSIR-NET Mathematical Sciences tests, examples and also practice Mathematics
tests
|
556 videos|198 docs
|
Download as PDF
|
Explore Courses for Mathematics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Objective type Questions
,Previous Year Questions with Solutions
,Extra Questions
,GATE
,past year papers
,video lectures
,MCQs
,study material
,shortcuts and tricks
,GATE
,Subspace Topology - Topology
,Free
,UGC NET
,Sample Paper
,CSIR NET
,CSIR NET
,practice quizzes
,ppt
,CSIR NET
,Viva Questions
,Subspace Topology - Topology
,Important questions
,UGC NET
,Subspace Topology - Topology
,GATE
,CSIR-NET Mathematical Sciences | Mathematics for IIT JAM
,Summary
,UGC NET
,mock tests for examination
,Semester Notes
,Exam
,CSIR-NET Mathematical Sciences | Mathematics for IIT JAM
,CSIR-NET Mathematical Sciences | Mathematics for IIT JAM
,
Additional Information about Subspace Topology - Topology, CSIR-NET Mathematical Sciences for Mathematics Preparation
Subspace Topology - Topology, CSIR-NET Mathematical Sciences Free PDF Download
The Subspace Topology - Topology, CSIR-NET Mathematical Sciences is an invaluable resource that delves deep into the core of the Mathematics exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Subspace Topology - Topology, CSIR-NET Mathematical Sciences now and kickstart your journey towards success in the Mathematics exam.
Importance of Subspace Topology - Topology, CSIR-NET Mathematical Sciences
The importance of Subspace Topology - Topology, CSIR-NET Mathematical Sciences cannot be overstated, especially for Mathematics aspirants.
This document holds the key to success in the Mathematics exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Subspace Topology - Topology, CSIR-NET Mathematical Sciences Notes
Subspace Topology - Topology, CSIR-NET Mathematical Sciences Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Subspace Topology - Topology, CSIR-NET Mathematical Sciences.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Subspace Topology - Topology, CSIR-NET Mathematical Sciences Notes on EduRev are your ultimate resource for success.
Subspace Topology - Topology, CSIR-NET Mathematical Sciences Mathematics Questions
The "Subspace Topology - Topology, CSIR-NET Mathematical Sciences Mathematics Questions" guide is a valuable resource for all aspiring students preparing for the
Mathematics exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Subspace Topology - Topology, CSIR-NET Mathematical Sciences on the App
Students of Mathematics can study Subspace Topology - Topology, CSIR-NET Mathematical Sciences alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Subspace Topology - Topology, CSIR-NET Mathematical Sciences,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Subspace Topology - Topology, CSIR-NET Mathematical Sciences is prepared as per the latest Mathematics syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup