UDC 513.83

MATHEMATICS

V. V. PROIZVOLOV

ON HEREDITARY AND COLLECTIVE NORMALITY OF A PERIPHERALLY BICOMPACT TREE-LIKE SPACE

(Presented by Academician P. S. Aleksandrov, 28 I 1970)

As numerous and simple examples show, a peripherally bicompact Hausdorff space need not be normal. Such a space, as Freudenthal showed (^1), is only automatically completely regular. However, a tree-like* peripherally bicompact space is normal, as is proved here. Moreover, the following theorems are valid.

Theorem 1. Any two disjoint closed sets of a tree-like peripherally bicompact space are separated by a discrete set.

Proof. In other words, it is necessary to prove that if (A, B \subset X) are closed, where (X) is peripherally bicompact and tree-like, (A \cap B = \Lambda), then there exists a discrete set (C \subset X) such that (X \setminus C = X_1 \cup X_2), (A \subset X_1), (B \subset X_2), where (X_1) and (X_2) are open in (X), (X_1 \cap X_2 = \Lambda).

Any two points in a tree-like peripherally bicompact space are joined by a unique ordered continuum (^2). If (D \subset V \subseteq X), then by the linear hull of the set (D) in the set (V) we shall mean the union of all those ordered continua which join points of (D) pairwise and are entirely contained in (V).

Lemma 1. Let (D \subset V \subseteq X), where (D) is closed, (V) is open, and (X) is a tree-like peripherally bicompact space. Then the linear hull (LD) of the set (D) in the set (V) is closed in (X).

We prove the lemma. Suppose that (x \in [LD] \setminus LD). Take a connected neighborhood (Ox) with finite boundary such that (Ox \cap D = \Lambda). Since (x \in [LD] \setminus LD), there exists an infinite number of ordered continua entirely belonging to (LD \cap [Ox]) and joining pairwise boundary points of (Ox). But the boundary of (Ox) is finite, and among these continua two distinct ones will be found joining a pair of boundary points of (Ox), which contradicts Lemma 1 of (^2).

Lemma 2. The boundary of a connected open subset in a tree-like peripherally bicompact space is punctiform.

Lemma 3. If a point (c) separates a connected subset (D) of a tree-like peripherally bicompact space (X) between points (a) and (b), (a, b \in D), then the point (c) also separates any subset containing (D) between the same points.

Returning to the proof of Theorem 1, take the linear hull (LA) of the set (A) in the set (X \setminus B) and then the linear hull (LB) of the set (B) in (X \setminus LA). We decompose the locally connected set (\widetilde X = X \setminus (LA \cup LB)) into open components: (\widetilde X = \bigcup_{\alpha} \widetilde X_\alpha). It is further asserted that every ([\widetilde X_\alpha]) contains not more than one point of (LA) and (LB).

Indeed, if, for example, (a_1, a_2 \in [\widetilde X] \cap LA), then two possibilities arise. The first: the ordered continuum (L), joining the points

* A space is tree-like if it is connected and if any two points in it are separated by a third. It is automatically Hausdorff.

(a_1) and (a_2), and lying entirely in ([\widetilde X_\alpha]), such that (L\cap LB=\Lambda). But then there is a contradiction with Lemma 2. Second: suppose there is a point (b\in L\cap LB). But then, by Lemma 3, the point (b) separates ([\widetilde X_\alpha]) between (a_1) and (a_2), and also, as is easy to conclude, (b) separates (\widetilde X_\alpha), which contradicts the connectedness of (\widetilde X_\alpha).

Next, if ([\widetilde X_\alpha]\setminus \widetilde X_\alpha) consists of two points, then we take a point (c_\alpha\in \widetilde X_\alpha) separating ([\widetilde X_\alpha]) between these two points. The union of all such points (c_\alpha) forms a discrete set (C) closed in (X).

The locally connected (X\setminus C) is decomposed into open components; those of them which intersect (LA) are united—we obtain (OA); those which intersect (LB) are united—we obtain (OB). There is no component which would intersect both (LA) and (LB). Suppose the contrary: suppose there is a component (P) of the set (X\setminus C) such that (P\cap LA\ne \Lambda) and (P\cap LB\ne \Lambda). But then there is a component (Q) of the set (P\setminus (LA\cup LB)) such that ([Q]) intersects both (LA) and (LB), in view of the connectedness of (P). There is a component (\widetilde X_\alpha) of the set (X\setminus (LA\cup LB)) containing (Q). From the latter it follows that in the set (C) there is a point (c_\alpha) separating ([Q]) between the points
[
a=[Q]\cap LA
]
and
[
b=[Q]\cap LB,
]
and moreover (c_\alpha\in Q), but this gives a contradiction. Thus the theorem is proved.

Corollary 1. The large inductive dimension of a dendroidal peripherally bicompact space is equal to one.*

In addition, Theorem 1 implies the normality of a predendroidal peripherally bicompact space. But a stronger assertion is true.

Theorem 2. A dendroidal peripherally bicompact space is hereditarily normal.

Proof. Let (\widetilde X\subseteq X) be a subspace of the dendroidal peripherally bicompact space (X), and let (A,B\subset \widetilde X) be closed in (\widetilde X) and (A\cap B=\Lambda). Consider the set (C=[A]\cap [B]). Obviously, (C\subseteq X\setminus \widetilde X). We decompose the locally connected open set (X\setminus C) in (X) into open components:
[
X\setminus C=\bigcup_\alpha X_\alpha .
]
Each (X_\alpha) is a peripherally bicompact dendroidal subspace of (X). Indeed, every subset of a peripherally bicompact dendroidal space is itself peripherally bicompact, since in a peripherally bicompact dendroidal space every point has arbitrarily small neighborhoods with finite boundaries ((^3)). Therefore, by Theorem 1, for the sets (A_\alpha=A'\cap X_\alpha) and (B_\alpha=B'\cap X_\alpha), where (A'=[A]\setminus C) and (B'=[B]\setminus C), there exist disjoint neighborhoods (OA_\alpha) and (OB_\alpha) such that (OA_\alpha,OB_\alpha\subset X_\alpha). Let us also note that (A\subseteq A') and (B\subseteq B').

It is not difficult to see that
[
OA=\bigcup_\alpha OA_\alpha
]
and
[
OB=\bigcup_\alpha OB_\alpha
]
are disjoint neighborhoods of the sets (A) and (B). This means that (\widetilde X) is normal, and Theorem 2 is proved.

Corollary 2. A dyadic dendroidal bicompactum is metrizable.

We had put forward a hypothesis on the metrizability of a hereditarily normal dyadic bicompactum, which B. A. Efimov succeeded in proving ((^4)). In combination with Theorem 2 this gives Corollary 2.

Theorem 3. A dendroidal peripherally bicompact space is collectively normal.

Lemma 4. The set (LD) mentioned in Lemma 1 is locally connected.

Let us prove the lemma. Let (x\in LD) be a point and let (Ox) be its connected neighborhood in (X) such that (Ox\subset V). It is asserted that the set (O_1x=Ox\cap LD), which is a neighborhood of the point (x) in the set (LD), is connected. Indeed, every point (y\in O_1x) can be joined to the point (x) by an ordered continuum lying entirely in (Ox), and hence also lying in (O_1x) in the sense of the linear hull. The lemma is proved.

* The question of (\operatorname{Ind}) of a dendroidal peripherally bicompact space was raised by B. A. Pasynkov.

Lemma 5. Let (\Gamma \subset X) be a discrete subset of the tree-like peripherally bicompact space (X), and suppose that (X \setminus \Gamma) is connected. Then there exists a closed set (C \subset X) such that (X \setminus C) is decomposed into open components, each of which contains at most one point of (\Gamma).

We prove the lemma. Let (\Gamma={x_\alpha}); for an arbitrary (x_\alpha \in \Gamma) denote (\Gamma_\alpha=\Gamma\setminus x_\alpha). The linear hull (L\Gamma_\alpha) of the set (\Gamma_\alpha) in (X) is closed by Lemma 1, is connected, and (x_\alpha \notin L\Gamma_\alpha). The latter follows from the fact that (X\setminus x_\alpha) is connected by the hypothesis of the lemma, and any two points in (X\setminus x_\alpha) are joined by a unique ordered continuum (see Lemma 1 from ((^2))). By (V_\alpha) we denote the open component of the set (X\setminus L\Gamma_\alpha) containing the point (x_\alpha). It is asserted that (V_{\alpha_1}\cap V_{\alpha_2}=\Lambda). Suppose the contrary: there is a point (c\in V_{\alpha_1}\cap V_{\alpha_2}). Then (V_{\alpha_1}\cup V_{\alpha_2}) is connected. We note that ([V_{\alpha_1}]\setminus V_{\alpha_1}) consists of a single point; denote it by (y_{\alpha_1}). This latter circumstance can be justified with the help of Lemma 3. Correspondingly, by (y_{\alpha_2}) denote the point ([V_{\alpha_2}]\setminus V_{\alpha_2}). From what has been said it is clear that

[
[V_{\alpha_1}\cup V_{\alpha_2}]\setminus (V_{\alpha_1}\cup V_{\alpha_2})=y_{\alpha_1}\cup y_{\alpha_2}
]

and (V=[V_{\alpha_1}\cup V_{\alpha_2}]) is a tree-like peripherally bicompact subspace of the space (X). We note that (y_{\alpha_2}\notin V_{\alpha_1}). Indeed, if (y_{\alpha_2}\in V_{\alpha_1}), then we would join (x_{\alpha_2}) with (y_{\alpha_1}) by an ordered continuum, which necessarily would pass through (y_{\alpha_2}), and then join the point (y_{\alpha_1}) with (x_\beta\in\Gamma\setminus (x_{\alpha_1},x_{\alpha_2})); consequently, we would obtain that (x_{\alpha_2}) and (x_\beta) are joined by an ordered continuum to which the point (y_{\alpha_2}) belongs, which means: (y_{\alpha_2}\in L\Gamma_{\alpha_1}). But the latter at the same time means that (y_{\alpha_2}\notin V_{\alpha_1}). Accordingly, (y_{\alpha_1}\notin V_{\alpha_2}). However, the point (y_{\alpha_1}\in L\Gamma_{\alpha_1}) and the point (y_{\alpha_2}\in L\Gamma_{\alpha_1}), for, joining (x_{\alpha_2}) and (x_\beta) by an ordered continuum, we see that it passes through (y_{\alpha_2}). The ordered continuum (L_1), joining (x_\beta) with (y_{\alpha_1}), does not intersect (V_{\alpha_1}), since (x_\beta,y_{\alpha_1}\in L\Gamma_{\alpha_1}), and, moreover, (L_1\cap V_{\alpha_2}=\Lambda), since the continuum joining (x_\beta) with (x_{\alpha_1}) belongs entirely to (L\Gamma_{\alpha_2}) and contains (L_1) in itself. Thus,

[
L_1\cap (V_{\alpha_1}\cup V_{\alpha_2})=\Lambda.
]

Similarly,

[
L_2\cap (V_{\alpha_1}\cup V_{\alpha_2})=\Lambda,
]

where (L_2) is the ordered continuum joining the points (x_\beta) and (y_{\alpha_2}). Denote by (L) the connected closed set (L_1\cup L_2). It is seen that the connected closed sets (L) and ([V]) intersect in two points (y_{\alpha_1}) and (y_{\alpha_2}), which contradicts the fact that in a tree-like peripherally bicompact space the intersection of connected closed sets is empty or connected (see Lemma 2 from ((^2))). Hence, (V_{\alpha_1}\cap V_{\alpha_2}=\Lambda), and as the set (C) we take (X\setminus \bigcup V_\alpha). Lemma 5 is proved.

Proof of Theorem 3. Let ({A_\alpha}) be a discrete family of closed subsets of the tree-like peripherally bicompact space (X). By transfinite induction we extend each (A_\alpha) to some closed locally connected (\widetilde A_\alpha) in the following way. The set

[
\widetilde A_1=LA_1
]

in the set

[
X\setminus \bigcup_{\alpha\ne 1} A_\alpha.
]

If all (\widetilde A_\alpha) have been constructed for all (\alpha<\beta), then (\widetilde A_\beta) is defined as follows: (\widetilde A_\beta) is the union of all those ordered continua joining pairs of points of (A_\beta) which do not intersect the closed set

[
\bigcup_{\alpha\le \beta}\widetilde A_\alpha \cup \bigcup_{\alpha\ne \beta} A_\alpha.
]

In order that the inclusion (A_\alpha\subset \widetilde A_\alpha) hold, by definition we adjoin to (\widetilde A_\alpha) all points of (A_\alpha) itself.

By Lemmas 1 and 4, each (\widetilde A_\alpha) is a locally connected closed set in (X). Decompose (\widetilde A_\alpha) into components; we obtain a discrete system of closed sets:

[
\widetilde A_\alpha=\bigcup_\gamma \widetilde A_{\alpha\gamma}.
]

The system ({\widetilde A_{\alpha\gamma}}), over all admissible (\alpha) and (\gamma), is a discrete system of connected closed sets.

The subspace

[
\widetilde X=X\setminus \bigcup_{\alpha,\gamma}\widetilde A_{\alpha\gamma}
]

is decomposed into open components:

[
\widetilde X=\bigcup_\beta \widetilde X_\beta.
]

At the same time, ([\widetilde X_\beta]) for any (\beta) contains at most one point from each (\widetilde A_{\alpha\gamma}); this means that

[
\Gamma_\beta=[\widetilde X_\beta]\setminus \widetilde X_\beta
]

is a discrete set. Applying Lemma 5 to the subspace ([\widetilde X_\beta]) and its discrete subset (\Gamma_\beta), we obtain a closed set (C_\beta), (C_\beta\subset \widetilde X_\beta). It is possible

to show that the set (C=\bigcup_{\beta} C_\beta) is closed. Suppose the contrary: there is a point (x\in [C]\setminus C). Take an arbitrary neighborhood (O_x) of it with finite boundary. The neighborhood (O_x) intersects infinitely many sets of the form (\widetilde X_\beta), but since the boundary of (O_x) is finite and the family ({\widetilde X_\beta}) is disjoint, this means that infinitely many sets of the form (\widetilde X_\beta) are entirely contained in (O_x). Thus every neighborhood of the point (x) with finite boundary contains infinitely many sets of the form (\widetilde X_\beta) and, consequently, intersects infinitely many sets of the form (\widetilde A_\alpha), which contradicts the discreteness of the family ({\widetilde A_\alpha}). Hence (C) is closed. After this it is not hard to see that (X\setminus C) decomposes into open components, each of which contains at most one set of the form (\widetilde A_\alpha); from this it follows at once that the discrete family ({A_\alpha}) is placed in a system of pairwise disjoint neighborhoods. Consequently, (X) is collectively normal. The theorem is proved.

Example. There exist dendroid spaces that are not normal, as the following example shows*. On the half-plane we introduce a new topology as follows. If (x\in L), where (L) is the boundary line, then a basic neighborhood (O_x) is an open disk tangent to (L) at the point (x), together with the point (x). If, however, (x\notin L), then its basic neighborhood is an arbitrary interval containing (x) and perpendicular to (L). The space obtained in this way is dendroid, completely regular, but not normal (compare it with the Niemytzki space!).

Faculty of Mechanics and Mathematics
M. V. Lomonosov Moscow State University

Received
26 XII 1970

References

1. H. Freudenthal, Ind. Math. Amst., 13, 184 (1951).
2. V. V. Prizvolov, DAN, 189, No. 4 (1969).
3. G. L. Gurin, Vestn. MGU, No. 1, 9 (1969).
4. B. A. Efimov, Tr. Mosk. matem. obshch., 14, 211 (1965).

* The example answers a question raised by O. V. Lokutsievskii.