# Alexandroff Double Circle

We discuss the Alexandroff double circle, which is a compact and non-metrizable space. A theorem about the hereditarily normality of a product space $Y_1 \times Y_2$ is also discussed.

Let $C_1$ and $C_2$ be the two concentric circles centered at the origin with radii 1 and 2, respectively. Specifically $C_i=\left\{(x,y) \in \mathbb{R}^2: x^2 + y^2 =i \right\}$ where $i=1,2$. Let $X=C_1 \cup C_2$. Furthermore let $f:C_1 \rightarrow C_2$ be the natural homeomorphism. Figure 1 below shows the underlying set.

_______________________________________________________________
Figure 1 – Underlying Set

_______________________________________________________________

We define a topology on $X$ as follows:

• Points in $C_2$ are isolated.
• For each $x \in C_1$ and for each positive integer $j$, let $O(x,j)$ be the open arc in $C_1$ whose center contains $x$ and has length $\frac{1}{j}$ (in the Euclidean topology on $C_1$). For each $x \in C_1$, an open neighborhood is of the form $B(x,j)$ where
$\text{ }$

$B(x,j)=O(x,j) \cup (f(O(x,j))-\left\{f(x) \right\}$).

The following figure shows an open neighborhood at point in $C_1$.

_______________________________________________________________
Figure 2 – Open Neighborhood

___________________________________________________________________________________

A List of Results

It can be verified that the open neighborhoods defined above form a base for a topology on $X$. We discuss the following points about the Alexandroff double circle.

1. $X$ is a Hausdorff space.
2. $X$ is not separable.
3. $X$ is not hereditarily Lindelof.
4. $X$ is compact.
5. $X$ is sequentially compact.
6. $X$ is not metrizable.
7. $X$ is not perfectly normal.
8. $X$ is completely normal (and thus hereditarily normal).
9. $X \times X$ is not hereditarily normal.

The proof that $X \times X$ is not hereditarily normal can be generalized. We discuss this theorem after presenting the proof of Result 9.
___________________________________________________________________________________

Results 1, 2, 3

It is clear that the Alexandroff double circle is a Hausdorff space. It is not separable since the outer circle $C_2$ consists of uncountably many singleton open subsets. For the same reason, $C_2$ is a non-Lindelof subspace, making the Alexandroff double circle not hereditarily Lindelof. $\blacksquare$

___________________________________________________________________________________

Result 4

The property that $X$ is compact is closely tied to the compactness of the inner circle $C_1$ in the Euclidean topology. Note that the subspace topology of the Alexandroff double circle on $C_1$ is simply the Euclidean topology. Let $\mathcal{U}$ be an open cover of $X$ consisting of open sets as defined above. Then there are finitely many basic open sets $B(x_1,j_1)$, $B(x_2,j_2)$, $\cdots$, $B(x_n,j_n)$ from $\mathcal{U}$ covering $C_1$. These open sets cover the entire space except for the points $f(x_1), f(x_2), \cdots,f(x_n)$, which can be covered by finitely many open sets in $\mathcal{U}$. $\blacksquare$

___________________________________________________________________________________

Result 5

A space $W$ is sequentially compact if every sequence of points of $W$ has a subsequence that converges to a point in $W$. The notion of sequentially compactness and compactness coincide for the class of metric spaces. However, in general these two notions are distinct.

The sequentially compactness of the Alexandroff double circle $X$ hinges on the sequentially compactness of $C_1$ and $C_2$ in the Euclidean topology. Let $\left\{x_n \right\}$ be a sequence of points in $X$. If the set $\left\{x_n: n=1,2,3,\cdots \right\}$ is a finite set, then $\left\{x_n: n>m \right\}$ is a constant sequence for some large enough integer $m$. So assume that $A=\left\{x_n: n=1,2,3,\cdots \right\}$ is an infinite set. Either $A \cap C_1$ is infinite or $A \cap C_2$ is infinite. If $A \cap C_1$ is infinite, then some subsequence of $\left\{x_n \right\}$ converges in $C_1$ in the Euclidean topology (hence in the Alexandroff double circle topology). If $A \cap C_2$ is infinite, then some subsequence of $\left\{x_n \right\}$ converges to $x \in C_2$ in the Euclidean topology. Then this same subsequence converges to $f^{-1}(x)$ in the Alexandroff double circle topology. $\blacksquare$

___________________________________________________________________________________

Result 6

Note that any compact metrizable space satisfies a long list of properties, which include separable, Lindelof, hereditarily Lindelof. $\blacksquare$

___________________________________________________________________________________

Result 7

A space is perfectly normal if it is normal with the additional property that every closed set is a $G_\delta$-set. For the Alexandroff double circle, the inner circle $C_1$ is not a $G_\delta$-set, or equivalently the outer circle $C_2$ is not an $F_\sigma$-set. To see this, suppose that $C_2$ is the union of countably many sets, we show that the closure of at least one of the sets goes across to the inner circle $C_1$. Let $C_2=\bigcup \limits_{i=1}^\infty T_n$. At least one of the sets is uncountable. Let $T_j$ be one such. Consider $f^{-1}(T_j)$, which is also uncountable and has a limit point in $C_1$ (in the Euclidean topology). Let $t$ be one such point (i.e. every Euclidean open set containing $t$ contains points of $f^{-1}(T_j)$). Then the point $t$ is a member of the closure of $T_j$ (Alexandroff double circle topology). $\blacksquare$

___________________________________________________________________________________

Result 8

We first discuss the notion of separated sets. Let $T$ be a Hausdorff space. Let $E \subset T$ and $F \subset T$. The sets $E$ and $F$ are said to be separated (are separated sets) if $E \cap \overline{F}=\varnothing$ and $F \cap \overline{E}=\varnothing$. In other words, two sets are separated if each one does not meet the closure of the other set. In particular, any two disjoint closed sets are separated. The space $T$ is said to be completely normal if $T$ satisfies the property that for any two sets $E$ and $F$ that are separated, there are disjoint open sets $U$ and $V$ with $E \subset U$ and $F \subset V$. Thus completely normality implies normality.

It is a well know fact that if a space is completely normal, it is hereditarily normal (actually the two notions are equivalent). Note that any metric space is completely normal. In particular, any Euclidean space is completely normal.

To show that the Alexandroff double circle $X$ is completely normal, let $E \subset X$ and $F \subset X$ be separated sets. Thus we have $E \cap \overline{F}=\varnothing$ and $F \cap \overline{E}=\varnothing$. Note that $E \cap C_1$ and $F \cap C_1$ are separated sets in the Euclidean space $C_1$. Let $G_1$ and $G_2$ be disjoint Euclidean open subsets of $C_1$ with $E \cap C_1 \subset G_1$ and $F \cap C_1 \subset G_2$.

For each $x \in E \cap C_1$, choose open $U_x$ (Alexandroff double circle open) with $x \in U_x$, $U_x \cap C_1 \subset G_1$ and $U_x \cap \overline{F}=\varnothing$. Likewise, for each $y \in F \cap C_1$, choose open $V_y$ (Alexandroff double circle open) with $y \in V_y$, $V_y \cap C_1 \subset G_2$ and $V_y \cap \overline{E}=\varnothing$. Then let $U$ and $V$ be defined by the following:

$U=\biggl(\bigcup \limits_{x \in E \cap C_1} U_x \biggr) \cup \biggl(E \cap C_2 \biggr)$

$\text{ }$

$V= \biggl(\bigcup \limits_{y \in F \cap C_1} V_y \biggr) \cup \biggl(F \cap C_2 \biggr)$

Because $G_1 \cap G_2 =\varnothing$, the open sets $U_x$ and $V_y$ are disjoint. As a result, $U$ and $V$ are disjoint open sets in the Alexandroff double circle with $E \subset U$ and $F \subset V$.

For the sake of completeness, we show that any completely normal space is hereditarily normal. Let $T$ be completely normal. Let $Y \subset T$. Let $H \subset Y$ and $K \subset Y$ be disjoint closed subsets of $Y$. Then in the space $T$, $H$ and $K$ are separated. Note that $H \cap cl_T(K)=\varnothing$ and $K \cap cl_T(H)=\varnothing$ (where $cl_T$ gives the closure in $T$). Then there are disjoint open subsets $O_1$ and $O_2$ of $T$ such that $H \subset O_1$ and $K \subset O_2$. Now, $O_1 \cap Y$ and $O_2 \cap Y$ are disjoint open sets in $Y$ such that $H \subset O_1 \cap Y$ and $K \subset O_2 \cap Y$.

Thus we have established that the Alexandroff double circle is hereditarily normal. $\blacksquare$

For the proof that a space is completely normal if and only if it is hereditarily normal, see Theorem 2.1.7 in page 69 of [1],
___________________________________________________________________________________

Result 9

We produce a subspace $Y \subset X \times X$ that is not normal. To this end, let $D=\left\{d_n:n=1,2,3,\cdots \right\}$ be a countable subset of $X$ such that $\overline{D}-D\ne \varnothing$. Let $y \in \overline{D}-D$. Let $Y=X \times X-C_1 \times \left\{y \right\}$. We show that $Y$ is not normal.

Let $H=C_1 \times (X-\left\{y \right\})$ and $K=C_2 \times \left\{y \right\}$. These are two disjoint closed sets in $Y$. Let $U$ and $V$ be open in $Y$ such that $H \subset U$ and $K \subset V$. We show that $U \cap V \ne \varnothing$.

For each integer $j$, let $U_j=\left\{x \in X: (x,d_j) \in U \right\}$. We claim that each $U_j$ is open in $X$. To see this, pick $x \in U_j$. We know $(x,d_j) \in U$. There exist open $A$ and $B$ (open in $X$) such that $(x,d_j) \in A \times B \subset U$. It is clear that $x \in A \subset U_j$. Thus each $U_j$ is open.

Furthermore, we have $C_1 \subset U_j$ for each $j$. Based in Result 7, $C_1$ is not a $G_\delta$-set. So we have $C_1 \subset \bigcap \limits_{j=1}^\infty U_j$ but $C_1 \ne \bigcap \limits_{j=1}^\infty U_j$. There exists $t \in \bigcap \limits_{j=1}^\infty U_j$ but $t \notin C_1$. Thus $t \in C_2$ and $\left\{t \right\}$ is open.

Since $(t,y) \in K$, we have $(t,y) \in V$. Choose an open neighborhood $B(y,k)$ of $y$ such that $\left\{t \right\} \times B(y,k) \subset V$. since $y \in \overline{D}$, there exists some $d_j$ such that $(t,d_j) \in \left\{t \right\} \times B(y,k)$. Hence $(t,d_j) \in V$. Since $t \in U_j$, $(t,d_j) \in U$. Thus $U \cap V \ne \varnothing$. $\blacksquare$

___________________________________________________________________________________

Generalizing the Proof of Result 9

The proof of Result 9 requires that one of the factors has a countable set that is not discrete and the other factor has a closed set that is not a $G_\delta$-set. Once these two requirements are in place, we can walk through the same proof and show that the cross product is not hereditarily normal. Thus, the statement that is proved in Result 9 is the following.

Theorem
If $Y_1$ has a countable subset that is not closed and discrete and if $Y_2$ has a closed set that is not a $G_\delta$-set then $Y_1 \times Y_2$ has a subspace that is not normal.

The theorem can be restated as:

Theorem
If $Y_1 \times Y_2$ is hereditarily normal, then either every countable subset of $Y_1$ is closed and discrete or $Y_2$ is perfectly normal.

The above theorem is due to Katetov and can be found in [2]. It shows that the hereditarily normality of a cross product imposes quite strong restrictions on the factors. As a quick example, if both $Y_1$ and $Y_2$ are compact, for $Y_1 \times Y_2$ to be hereditarily normal, both $Y_1$ and $Y_2$ must be perfectly normal.

Another example. Let $W=\omega_1+1$, the succesor of the first uncountable ordinal with the order topology. Note that $W$ is not perfectly normal since the point $\omega_1$ is not a $G_\delta$ point. Then for any compact space $Y$, $W \times Y$ is not hereditarily normal. Let $C=\omega+1$, the successor of the first infinite ordinal with the order topology (essentially a convergent sequence with the limit point). The product $W \times C$ is the Tychonoff plank and based on the discussion here is not hereditarily normal. Usually the Tychonoff plank is shown to be not hereditarily normal by removing the cornor point $(\omega_1,\omega)$. The resulting space is the deleted Tychonoff plank and is not normal (see The Tychonoff Plank).

___________________________________________________________________________________

Reference

1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
2. Przymusinski, T. C., Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 781-826, 1984.
3. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

___________________________________________________________________________________

$\copyright \ \ 2012$