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Limit point(Cluster point) of the Singleton

Amer

Active member
Mar 1, 2012
275
Hey,
What is the limit point of the singleton ?
for example {x} in any topology
{x}' = phi
suppose not there exist x in {x}'
let U be an open set containing x, U should contain another point of {x} other than x but that impossible
did i do something wrong ?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
Can you supply some additional information?

For starters what you mean by {x}' = phi?
What's the quote?
And what do you mean by phi?
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
I think that if your topology satisfies the $T_1$ or $T_2$ axioms we can conclude that it is itself, because it is going to be a closed set and therefore it contains all its limit points, but I am not sure.

My 2 cents. (Wave)
 

Amer

Active member
Mar 1, 2012
275
Can you supply some additional information?

For starters what you mean by {x}' = phi?
What's the quote?
And what do you mean by phi?
Ok
The question is we call a Topological space X a TD space, if for all [tex]x \in X [/tex]
the derive set ( the set of all cluster points) of [tex]\{ x \} [/tex] is closed.
Prove that if a space is [tex]T_1[/tex] then it is a TD space and if a space is TD then it is [tex] T_0[/tex]

so I tried to understand the TD space, I found that each space is TD space
[tex](X,T) [/tex] be a topological space
let [tex] x \in X [/tex]
I want to prove that the derive set of the Singleton is closed, I get that the derive set is [tex]\phi[/tex] which means the set [tex]\{x\}[/tex] dose not have any cluster point !! is that true ?
 

Amer

Active member
Mar 1, 2012
275
I found my mistake, in any topology [tex](X,T)[/tex]
the derive set of [tex]\{x\}[/tex] dose not contains x but it maybe contain another element for example
[tex] X = \{a,b\} [/tex] and [tex]T = \{X,\phi , \{a\} \} [/tex]
the derive set of [tex]\{a\}[/tex] contains b
since each open set containing "b" contains "a" ( in fact we have just one open set containing b, X )

I am sorry because I am not clear in asking the question, but when I am writing I am thinking that all my thoughts are written or the reader has the same thoughts

Let [tex] (X,T) [/tex] be a TD space, and x,y in X
the set [tex] \{x \}' [/tex] is closed and dose not contain x
if [tex] y \in \{x\}' [/tex] , then [tex] U = X - \{x\}' [/tex] an open set which contains "x" and dose not contain y
if [tex] y [/tex] is not in [tex]\{x\}' [/tex] then there exist an open set [tex] U [/tex] containing "y" which dose not contains x ( the intersection of [tex] U- \{y\} \cap \{x\} = \phi [/tex] )
so the space is [tex]T_0[/tex] space.