1

Let $X$ be a compact connected Hausdorff space and $M$ be the set of compact connected subspaces of $X$.

Let $\mathscr{C}$ be a nonempty chain in $(M,\subset)$.

Let $a:\mathscr{C}\rightarrow \bigcup \mathscr{C}$ be a choice function.

How do I prove that there exists a point $x$ in $X$ such that for every neighborhood $U$ of $x$ in $X$, there exists $Y_0\in \mathscr{C}$ such that $x\in Y$ for all $Y\in \mathscr{C}$ with $Y\subset Y_0$?

Martin Sleziak
  • 56,060
user277190
  • 59
  • 1
    Why do you introduce the choice function $a$ and the neighborhood $U$, if they don't appear in the statement? – Stefan Hamcke Oct 05 '15 at 15:51
  • You might have made a mistake when formulating the question. (As previous comment says, you have introduced several things which you do not use at all.) But in the current formulation, you could probably use the fact that $\bigcap\mathscr{C}$ is non-empty. (An intersection of closed subsets of subsets which has finite intersection property is non-empty in compact spaces.) See also this question. – Martin Sleziak Oct 05 '15 at 16:19
  • Since you mentioned nets in the title, perhaps the intended question might have been something along the lines whether the net $(a(Y))_{Y\in\mathscr{C}}$ converges to some $x\in X$? – Martin Sleziak Oct 05 '15 at 16:20
  • If this is an exercise from some textbook or part of a larger proof from some book or paper, adding the reference might be useful for the user trying to answer your question. – Martin Sleziak Oct 05 '15 at 16:22

0 Answers0