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open sets such that [tex]\overline{U_n} \bigcap \overline{U_m}[/tex] for [tex]n \neq m [/tex]

and [tex] U_m \bigcap A \neq \phi [/tex] for all m (Use induction )

My work

I will work on [tex]x_1,x_2 ,...[/tex] subset of A

for k=2 let [tex]x_1 , x_2 \in A [/tex] there exist two disjoint open sets [tex]U_1 ,U_2 [/tex] such that [tex]x_1 \in U_1 \; , \; x_2 \in U_2 [/tex]

from regularity we can find two open sets [tex]V_1 , V_2 [/tex] such that

[tex]x_1 \in V_1 \subseteq \overline{V_1 } \subseteq U_1 [/tex]

[tex]x_2\in V_2 \subseteq \overline{V_2 } \subseteq U_2 [/tex]

and [tex]V_1 , V_2 [/tex] have the properties we are searching for

now suppose it is true for k now want to test if it is true for k+1

[tex]x_1 , x_2 ,...,x_{k+1} [/tex]

there exist two disjoint open sets [tex]U_n , U_m [/tex] such that [tex]x_n \in U_n [/tex] and [tex]x_m \in U_m [/tex] fixed these open sets

we have now for any [tex]x_i[/tex] of these x's k+1 open sets containing [tex]x_i [/tex] wihtout containing other x, Let

[tex]V_1 = \bigcap_i^{k+1} U_{i} [/tex] such that [tex]U_1 ,...,U_{k+1} [/tex] containing [tex]x_1 [/tex]

do the same for all x's we have now we use the regularity property we have [tex]H_1 \subseteq \overline {H_1} \subseteq V_1 [/tex]

these [tex]H_1,H_2 ,...,H_{k+1} [/tex] are the U's we are looking for

how about my proof any notes

Thanks