Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $R$ be Real line and let $F[R]$ be $\{x\subset R:\text{is finite}\}$ with Pixley-Roy topology.

Definition of Pixley-Roy topology is this: Basic neighborhoods of $F\in F[R]$ are the sets $$[F,V]=\{H\in F[R]; F\subseteq H\subseteq V\}$$ for open sets $V\supseteq F$, see e.g. here.

I'm not very clear to the topology of this space. I'm try to have an intuitional impress of the space. Could someone draw a picture for me such that help me to familar with this special toplogical space?

Thanks ahead:)

share|cite|improve this question
up vote 5 down vote accepted

Consider the very rough sketch below:

                p        s       q  t          x            r   u  
                            U                                W

Let $V=U\cup W$, the union of the two open intervals shown in the sketch, and let $F=\{p,q,r\}$. Then $[F,V]$ is the collection of all finite subsets $H$ of $\Bbb R$ such that $F\subseteq H$ and $H\subseteq V$. For example, let $H=\{p,q,r,s,t,u\}$; then $F\subseteq H\subseteq V$, so $H\in[F,V]$. On the other hand, $K=\{p,q,s,t,u\}$ is not in $[F,V]$, because $F\nsubseteq K$ (since $r\notin K$), and $G=\{p,q,s,t,u,x\}\notin[F,V]$ because $G\nsubseteq V$ (since $x\notin V$).

Added: It’s not easy to give a good global picture of $X=\mathscr{F}[\Bbb R]$, but perhaps these observations will help. For $n\in\Bbb Z^+$ let $X_n=\{F\in\mathscr{F}[\Bbb R]:|F|=n\}$.

  1. Each $X_n$ is a discrete subspace of $X$. Suppose that $F\in X_n$, and let $V$ be an open set in $\Bbb R$. If $F\nsubseteq V$, then of course $[F,V]=\varnothing$, so assume that $F\subseteq V$. If $G\in[F,V]\cap X_n$, then $F\subseteq G\subseteq V$; but $|F|=|G|=n$, so $F\subseteq G$ iff $F=G$, and therefore $[F,V]=\{F\}$.

  2. For each $n\in\Bbb Z^+$ let $Y_n=\bigcup_{k\le n}X_k$, the set of points of $X$ that contain at most $n$ points of $\Bbb R$; then $Y_n$ is closed in $X$. For suppose that $F\in X\setminus Y_n$; then $|F|>n$, and $[F,\Bbb R]\cap Y_n\varnothing$, since every member of $[F,\Bbb R]$ contains $F$ and therefore has more than $n$ points of $\Bbb R$.

  3. For each $n\in\Bbb Z^+$, $X_n$ is a dense set of isolated points of $Y_n$. That the points of $X_n$ are isolated in $Y_n$ follows from (1) and (2). To see that they are dense in $Y_n$, let $F\in Y_n$ be arbitrary, and let $V$ be any open subset of $\Bbb R$ containing $F$. Clearly $|F|\le n$. If $|F|=n$, $F\in[F,V]\cap X_n$. If $|F|<n$, let $G$ be a set of $n-|F|$ points of $V\setminus F$; clearly $F\cup G\in[F,V]\cap X_n$.

  4. An easy extension of (3) is that the Cantor-Bendixson rank of $Y_n$ is $n$, and that in fact $Y_n^{(k)}=Y_{n-k}$ for $k\le n$, where $Y_0=\varnothing$.

  5. Let $[F,V]$ and $[G,W]$ be basic open sets in $X$. Then $$[F,V]\cap[G,W]=[F\cup G,V\cap W]\;,$$ which is empty iff $F\cup G\nsubseteq V\cap W$. $[F,V]\cup[G,W]$ doesn’t have a really simple description.

  6. It’s not too hard to use (5) and the second countability of $\Bbb R$ to show that $X$ is ccc: every collection of pairwise disjoint non-empty subsets of $X$ is countable.

  7. $X$ is metacompact. To see this, let $\mathscr{U}$ be an open cover of $X$. Without loss of generality assume that the members of $\mathscr{U}$ are basic open sets. For each $F\in X$ there is a $U_F=[G_F,V_F]\in\mathscr{U}$ such that $F\in U_F$. Let $R_F=[F,V_F]$; $G_F\subseteq F$, so $[F,V_F]\subseteq U_F$, and $\mathscr{R}=\{R_F:F\in X\}$ is an open refinement of $\mathscr{U}$. Let $G\in X$ be arbitrary; if $G\in R_F\in\mathscr{R}$, then $F\subseteq G$. $G$ has only finitely many subsets, so $G$ belongs to at most finitely many members of $\mathscr{R}$. Thus, $\mathscr{R}$ is a point-finite open refinement of $\mathscr{U}$, and $X$ is metacompact. (In fact $X$ is hereditarily metacompact, by a similar argument.)

share|cite|improve this answer
Thanks Brain for the local struct picture. Could you show me more on the global struct of the space? – Paul Jul 14 '12 at 3:54

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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