Question

@ikram002p

Answer 1

counter example for : infinite union of closed sets is closed

\[ \bigcup \left[ -1 + \frac{1}{n},~1-\frac{1}{n}\right]\]

\[n \in \mathbb{N}\]

Answer 2

need help interpreting it

Answer 3

in tau standard ?

Answer 4

whats tau standard ?

Answer 5

xD
so ur talking about closed interval not sets !?

Answer 6

Oh is there a difference

Answer 7

cant i treat a closed interval as a closed set ?

Answer 8

no , they are different

Answer 9

see this one is interval , there unin would be
(-1,1 ) which is open interval :)

Answer 10

in tau standard ( standard topology) for an indexed family that u gave
its also a counter example as i see :O

Answer 11

My book says "union of finite closed sets is closed" not infinite.

Answer 12

how did u get (-1, 1) ?

Answer 13

exactly loser this is what we are talking about :)

Answer 14

@ganeshie8 are you wanting to break down that thingy above you have there?

Answer 15

yes.. take unions of all the intervals for each n

Answer 16

ok so take the biggest integer that u can , 1/n will goes to zero but it will never be zero , so term -1+1/n will be near to -1 but will not be -1 so its open beside -1 same thing with 1+1/n

Answer 17

I see, you're consdiering inf and sup of the resulting union. interesting

Answer 18

and inf and sup wont be inside union because n is never infinity

Answer 19

yep , this is for infinite sets , but for infinite its clear they are also closed

Answer 20

I kindof get it. so "infinite union of closed sets need not be a closed set"

however "finite union of closed sets is always closed"

is it ?

Answer 21

\[[-1+\frac{1}{1},1-\frac{1}{1}] \cup [-1+\frac{1}{2},1-\frac{1}{2}] \cup [-1+\frac{1}{3},1-\frac{1}{3}] \cup ... \cup [-1+0,1-0] \\ [0,0] \cup [\frac{-1}{2},\frac{1}{2}] \cup [-\frac{2}{3},\frac{2}{3}] \cup .. \cup[-1,1]\]
I guess we would say that last interval is open because n is never infinity it only approaches infinity
(if you know what i mean)
(and i know it doesn't really make since to say n is never infinity because infinity isn't a number but I think you guys know what I mean)

Answer 22

sense*

Answer 23

@ganeshie8 pardon a bit , what do u mean by closed set ?

Answer 24

exactly ^ im thinking the exact same atm

Answer 25

i couldn't answer ur question sense closed set have other definition xD
so i wanna know what u are refers to

Answer 26

open set is a set whose limit points are outside the set

Answer 27

closed set is a set whose limit points are inside the set

Answer 28

basically thats what i understand from the online definitions