Skip to main content
MathOverflow

Return to Question

added 2 characters in body
Source Link
Rafał Gruszczyński
Rafał Gruszczyński
  • 1.1k
  • 10
  • 17

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}_x$ (the neighbourhood filter at $x$): $$\mathscr{F}=\{Q\in2^X\mid\exists_{V_1,\ldots,V_n\in\mathscr{O}_x}\,V_1\cap\ldots\cap V_n\subseteq Q\} $$$$\mathscr{F}=\{Q\in2^X\mid(\exists V\in\mathscr{O}_x)\,V\subseteq Q\} $$

What I am interested in is under what conditions (if any) can we guarantee existence of a chain $C\subseteq\mathscr{O}_x$ (may be uncountable) such that the filter $\mathscr{F}_C$ generated by $C$ is equal to $\mathscr{F}$.

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}_x$: $$\mathscr{F}=\{Q\in2^X\mid\exists_{V_1,\ldots,V_n\in\mathscr{O}_x}\,V_1\cap\ldots\cap V_n\subseteq Q\} $$

What I am interested in is under what conditions (if any) can we guarantee existence of a chain $C\subseteq\mathscr{O}_x$ (may be uncountable) such that the filter $\mathscr{F}_C$ generated by $C$ is equal to $\mathscr{F}$.

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}_x$ (the neighbourhood filter at $x$): $$\mathscr{F}=\{Q\in2^X\mid(\exists V\in\mathscr{O}_x)\,V\subseteq Q\} $$

What I am interested in is under what conditions (if any) can we guarantee existence of a chain $C\subseteq\mathscr{O}_x$ (may be uncountable) such that the filter $\mathscr{F}_C$ generated by $C$ is equal to $\mathscr{F}$.

Source Link
Rafał Gruszczyński
Rafał Gruszczyński
  • 1.1k
  • 10
  • 17

Convergent filters generated by (not necessarily countable) chains

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}_x$: $$\mathscr{F}=\{Q\in2^X\mid\exists_{V_1,\ldots,V_n\in\mathscr{O}_x}\,V_1\cap\ldots\cap V_n\subseteq Q\} $$

What I am interested in is under what conditions (if any) can we guarantee existence of a chain $C\subseteq\mathscr{O}_x$ (may be uncountable) such that the filter $\mathscr{F}_C$ generated by $C$ is equal to $\mathscr{F}$.