Answered: Explain with an example of a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Explain with an example of a topological space which is singleton that is open but not close and another one that is close but not open

Expert Solution

Step 1: topological space

Example in refer of not open but closed .

Take $X equals straight real numbers comma tau equals s tan d a r d space t o p log y space o n space straight real numbers space a n d space A equals open curly brackets 1 close curly brackets subset of or equal to X$ , Here A is singletone set which is closed but not open

closed because ,any outside point of A ,is not limit point of A in this reason that there must exists a nbd of that outside point of A that disjoint from A and not open because ,there is no open interval containing 1,subset of A.

Example in refer of open but not closed .

Take $X equals left curly bracket 1 comma 2 comma 3 right curly bracket space a n d space tau equals open curly brackets ϕ comma open curly brackets 1 close curly brackets comma open curly brackets 2 close curly brackets comma open curly brackets 1 comma 2 close curly brackets comma X close curly brackets$ ,Here $open parentheses X comma tau close parentheses$ is a topological space.

Now consider singletone set $A equals left curly bracket 1 right curly bracket subset of or equal to X$ .

Here $A space i s space o p e n space b u t space n o t space c l o s e d space i n space open parentheses X comma tau close parentheses$ .

because $3 not an element of A space comma b u t space space open square brackets X set minus open curly brackets 3 close curly brackets close square brackets intersection A not equal to ϕ$ .

here possible open set containing $3$ is only $X$ , so 3 is limit point of A ,but not in A .So A is not closed in $open parentheses X comma tau close parentheses$ .