4
$\begingroup$

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 $\mathscr{F}$.

$\endgroup$
0
4
$\begingroup$

Topological spaces where each point has a totally ordered local basis are known as $\textit{well-based}$ spaces. The notion of a well-based space is a generalization of the notion of a first countable space since the first countable spaces are the spaces where every point has a countable totally ordered local basis. A radial space is a topological space where $x\in\overline{A}$ iff there is some regular cardinal $\kappa$ and sequence $(a_{\alpha})_{\alpha<\kappa}$ of elements in $A$ where $(a_{\alpha})_{\alpha<\kappa}\rightarrow x$. Every well-based space is radial, and every radial space is the quotient of a well-based space. The papers 3,4 mention well-based spaces, but I have not found any other mathematical literature that mentions well-based spaces and calls these spaces well-based.

Every first countable space is clearly well-based. Furthermore, if $X$ is a totally ordered set, then $X$ is well based in the order topology and in the lower limit topology.

$\textbf{When the sets $\mathscr{O}_{x}$ have the same cofinality $\kappa$}$

The well-based spaces where each $\mathscr{O}_{x}$ has the same cofinality seem to have a special importance among all the well-based spaces.

Suppose that $\kappa$ is a regular cardinal. Then define a $\kappa$-well based space to be a well based space where each $\mathscr{O}_{x}$ has a cofinal chain of cofinality $\kappa$. Even though the notion of a $\kappa$-well based space seems like a natural notion to study, I have not found any papers that mention $\kappa$-well based spaces. The notion of a $\kappa$-well based space is the correct generalization of the notion of a separable space to $\kappa$-complete spaces.

Suppose that $\kappa$ is a regular cardinal. Then we say that a space $X$ is $\kappa$-complete if whenever $|I|<\kappa$ and $O_{i}$ is open for each $i\in I$, then $\bigcap_{i\in I}O_{i}$ is also open. Completely regular $\kappa$-complete spaces are also called $P_{\kappa}$-spaces.

If $X$ is a topological space and $x\in X$, then define the character $\chi(X,x)$ of $X$ at $x$ to be the smallest cardinality of a subset $R\subseteq\mathscr{O}_{x}$ that generates $\mathscr{O}_{x}$.

Clearly a topological space $X$ is $\kappa$-well based iff $X$ is a $\kappa$-complete space where either $x$ is isolated or $\chi(X,x)=\kappa$ for each $x\in X$. For example, the first countable spaces are precisely the $\kappa$-well based spaces.

For example, the spaces where each $\mathscr{O}_{x}$ is generated by a chain of cofinality $\aleph_{0}$ are precisely the first countable spaces. It turns out that the $\kappa$-complete spaces $X$ where each $\mathscr{O}_{x}$ is generated by a chain of cofinality $\kappa$ behave very similar to the first countable spaces. In fact, the basic theory of $\kappa$-complete spaces is very similar to the basic theory of topological spaces since basic results about topological spaces often generalize the results about $\kappa$-complete spaces, ad the notion of a $\kappa$-well based space is the correct way to generalize the notion of a first countable space to $\kappa$-complete spaces. The papers 1,2 develop some of the basic theory of $\kappa$-complete spaces and generalize some facts about topological spaces to $\kappa$-complete spaces.

$\kappa$-complete spaces can easily be generated by topological spaces. Suppose that $X$ is a topological space and $\kappa$ is a regular cardinal. Then define $(X)_{\kappa}$ to be the topological space with the same underlying set as $X$ but where $(X)_{\kappa}$ is generated by the basis consisting of sets of the form $\bigcap_{i\in I}U_{i}$ where each $U_{i}$ is open in $X$ and where $|I|<\kappa$. Then $(X)_{\kappa}$ is always a $\kappa$-complete space.

The following results are a rather straightforward generalizations of results about first countable spaces, but they illustrate that the notion of a $\kappa$-well based space is the correct generalization of the notion of a first countable space.

$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-well based space. If $A\subseteq X$, then $x\in\overline{A}$ if and only if there is some sequence $(a_{\alpha})_{\alpha<\kappa}$ with $a_{\alpha}\in A$ for each $\alpha<\kappa$ converging to $x$.

$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-well based space. Let $Y$ be a topological space and let $f:X\rightarrow Y$ be a function. Then the following are equivalent.

  1. $f:X\rightarrow Y$ is a continuous function.

  2. $f:X\rightarrow(Y)_{\kappa}$ is a continuous function.

  3. Whenever $(x_{\alpha})_{\alpha<\kappa}\rightarrow x$, we have $(f(x_{\alpha}))_{\alpha<\kappa}\rightarrow f(x).$

$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-complete space. Then the following are equivalent.

  1. Every open cover $\mathcal{U}$ of cardinality $\kappa$ has a subcover of cardinality less than $\kappa$.

  2. Every sequence $(x_{\alpha})_{\alpha<\kappa}$ accumulates at some point.

$\mathbf{Theorem}$ Let $X$ be a $\kappa$-well based space. If $(x_{\alpha})_{\alpha<\kappa}$ is a sequence that accumulates at some point $x$, then there exists a subsequence $(x_{j(\alpha)})_{\alpha<\kappa}$ where $j:\kappa\rightarrow\kappa$ is strictly increasing but where $x_{j(\alpha)}\rightarrow x$.

$\mathbf{Theorem}$ Suppose $X$ is a $\kappa$-well based space. Then the following are equivalent.

  1. If $\mathcal{U}$ is an open cover with cardinality $\kappa$, then $\mathcal{U}$ has a subcover of cardinality less than $\kappa$.

  2. If $(x_{\alpha})_{\alpha<\kappa}$ is a sequence, then there is some strictly increasing map $j:\kappa\rightarrow\kappa$ such that $(x_{j(\alpha)})_{\alpha<\kappa}$ converges to some point.

We therefore conclude that the $\kappa$-well based spaces are spaces that one can study simply by considering the $\kappa$-sequences and their convergence. Furthermore, one can get by with thinking about $\kappa$-well based spaces n the same way that one thinks about first countable spaces.

I hope I did not stray too far from the information which you are looking for in this answer.

$\mathbf{References}$

  1. $\omega_{\mu}$-additive topological spaces. Giuliano Artico; Roberto Moresco Rendiconti del Seminario Matematico della Università di Padova (1982) Volume: 67, page 131-141 ISSN: 0041-8994

  2. Remarks on some topological spaces of high power. Roman Sikorski Fundamenta Mathematicae (1950) Volume: 37, Issue: 1, page 125-136 ISSN: 0016-2736

  3. Convergence properties and compactifications. Robert Leek. University of Oxford

  4. An internal characterisation of radiality. Robert Leek. University of Oxford

$\endgroup$
1
  • $\begingroup$ Joseph, thanks a lot for your effort. I have to digest it now and then I will probably have some more questions. But basically this has already helped me partially solved a problem which motivated the question. $\endgroup$ – Mad Hatter Apr 9 '15 at 21:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.