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Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U}$, st $x \in U_1$ and $y \in U_2$ ), then there is a point $x\inX$ such that $|U \in \mathscr{U}: x \in U|>\omega$. Is there a countable pseudocharacter Hausdorff space X with the property A?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U}$, st $x \in U_1$ and $y \in U_2$ ), then there is a point $x\in X$ such that $|U \in \mathscr{U}: x \in U|>\omega$. Is there a countable pseudocharacter Hausdorff space X with the property A?

Let X be a countable pseudocharacter Hausdorff space and $\mathscr{U}$ is a given base of X.

Suppose that $\mathscr{U'}$ is any subcollection of $\mathscr{U}$ that witnesses Hausdorff property of X (= $\forall x,y \in X$, there

  exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U'}$, st $x \in U_1$ and $y \in U_2$ ).

Property A: if there is a point $x \in X$ such that $|u \in \mathscr{U'}: x \in u| > \omega$

Is there a countable pseudocharacter Hausdorff space X, such that any $\mathscr{U'}$ of $\mathscr{U}$ satisfies Property A?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U}$, st $x \in U_1$ and $y \in U_2$ ), then there is a point $x\inX$ such that $|U \in \mathscr{U}: x \in U|>\omega$. Is there a countable pseudocharacter Hausdorff space X with the property A?

Let X be a countable pseudocharacter Hausdorff space and $\mathscr{U}$ is a base of X.

Suppose that $\mathscr{U'}$ is a subcollection of $\mathscr{U}$, $\forall x,y \in X$, there

exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U'}$, st $x \in U_1$ and $y \in U_2$.

Property A: if there is a point $x \in X$ such that $|u \in \mathscr{U'}: x \in u| > \omega$

Is there a countable pseudocharacter Hausdorff space X, such that any  $\mathscr{U'}$ of $\mathscr{U}$ satisfies Property A?

Let X be a countable pseudocharacter Hausdorff space and $\mathscr{U}$ is a given base of X.

Suppose that $\mathscr{U'}$ is any subcollection of $\mathscr{U}$ that witnesses Hausdorff property of X (= $\forall x,y \in X$, there

exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U'}$, st $x \in U_1$ and $y \in U_2$ ).

Property A: if there is a point $x \in X$ such that $|u \in \mathscr{U'}: x \in u| > \omega$

Is there a countable pseudocharacter Hausdorff space X, such that any $\mathscr{U'}$ of $\mathscr{U}$ satisfies Property A?