If the space $X$ is completely regular, we Know that The collection {$intZ(f)$:$f$ is a continuous function from $X$ to the real numbers} is an open base for open subsets of the space $X$ (i.e. If for each element $x$ and each open set $U_x$ of $X$, there exist a continuous real valued function $f:X→R$ f:X→\mathbb{R}$such that$x∈intZ(f)⊆Z(f)⊆U_x)$. I have two questions about converse of this theorem. these questions are almost the same, but I think these are different. 1.If for each element$x$and each open set$U_x$of$X$, there exist a continuous real valued function$f:X→R$f:X→\mathbb{R}$ such that $x∈intZ(f)⊆U_x$, then $X$ is completely regular.

2.If for each element $x$ and each open set $U_x$ of $X$, there exist a continuous real valued function $f:X→R$ f:X→\mathbb{R}$such that$x∈intZ(f)⊆Z(f)⊆U_x$, then$X$is completely regular. I think these two claims have counterexamples and these conditions don't emply the complete regularity of$X$. 3 added 56 characters in body If the space X$X$is completely regular, we Know that The collection {int(Z[f]):f$intZ(f)$:$f$is a continuous function from X$X$to the real numbers} is an open base for open subsets of the space X$X$(i.e. If for each element x$x$and each open set U$U_x$of X,$X$, there exist a continuous real valued function f:X→R$f:X→R$such that x∈intZ(f)⊆Z(f)⊆U).$x∈intZ(f)⊆Z(f)⊆U_x)$. I have two questions about converse of this theorem. these questions are almost the same, but I think these are different. 1.If for each element x$x$and each open set U$U_x$of X,$X$, there exist a continuous real valued function f:X→R$f:X→R$such that x∈intZ(f)⊆U,$x∈intZ(f)⊆U_x$, then X$X$is completely regular. 2.If for each element x$x$and each open set U$U_x$of X,$X$, there exist a continuous real valued function f:X→R$f:X→R$such that x∈intZ(f)⊆Z(f)⊆U,$x∈intZ(f)⊆Z(f)⊆U_x$, then X$X$is completely regular. I think these two claims have counterexamples and these conditions don't emply the complete regularity of X.$X\$.

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If the space X is completely regular, we Know that The collection {int(Z[f]):f is a continuous function from X to the real numbers} is an open base for open subsets of the space X (i.e. If for each element x and each open set U of X, there exist a continuous real valued function f:X→R such that x∈intZ(f)⊆Z(f)⊆U). I have two questions about converse of this theorem. these questions are almost the same, but I think these are different.

1.If for each element x and each open set U of X, there exist a continuous real valued function f:X→R such that x∈intZ(f)⊆U, then X is completely regular.

2.If for each element x and each open set U of X, there exist a continuous real valued function f:X→R such that x∈intZ(f)⊆Z(f)⊆U, then X is completely regular.

I think these two claims have conterexampels counterexamples and these conditions don't emplies emply the complete regularity of X.

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