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- Mar 5, 2012

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For starters what you mean by {x}' = phi?

What's the quote?

And what do you mean by phi?

- Feb 29, 2012

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My 2 cents.

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Ok

For starters what you mean by {x}' = phi?

What's the quote?

And what do you mean by phi?

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 ?

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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.