MATHEMATICS

V. V. PROIZVOLOV

ON CONDENSATIONS ONTO EUCLIDEAN SPACES

(Presented by Academician P. S. Aleksandrov, March 12, 1963)

The main theorem of the present note is as follows:

Theorem 1. Let a condensation \(f:X\to E^n\) be given, where \(X\) is a connected, locally bicompact paracompact space, and \(E^n\) is Euclidean space. Then \(f\) is a homeomorphism.

For \(n=1\) this theorem has a strengthening:

Theorem 2. Let a condensation \(f:X\to L\) be given, where \(X\) is a connected and peripherally bicompact** space, and \(L\) is a line. Then \(f\) is a homeomorphism.

Proof of Theorem 2. In order that a condensation be a homeomorphism, it is necessary and sufficient that it be closed. Suppose that \(f\) is not closed. Then it is not hard to see that there will be a countable closed subset \(A\subset X\) such that \(fA\subset L\) is not closed. Let \(\xi=f^{-1}a\), where \(a\in [fA]\setminus fA\). The space \(X\) is Hausdorff, since it is condensed onto the Hausdorff space \(L\), and, being peripherally bicompact, it is completely regular \((^1)\). Therefore there exists at the point \(\xi\) a neighborhood \(O\xi\) with bicompact boundary \(F\) and such that \([O\xi]\cap A=\Lambda\), or, what is the same thing: \(f[O\xi]\cap fA=\Lambda\). There is an \(\varepsilon>0\) such that \(O_\varepsilon a\cap fF=\Lambda\).

The line \(L\) is representable as the sum of two rays: \(L=L^+\cup L^-\), \(L^+\cap L^-=a\). Essentially, two cases are logically possible:

1. \(a\in [(L^-\setminus a)\cap f[O\xi]]\), \(a\in [(L^+\setminus a)\cap f[O\xi]]\), and, in addition, for example, \(a\in [L^-\cap fA]\). In this case we proceed as follows. Choose a point \(b\) from \(L^-\) such that \(b\in fO\xi\) and \(|a-b|<\varepsilon\), where \(\varepsilon\) is the distance from the point \(a\) to \(fF\). Denote the segment \([a,b]\) by \(I\). The set \(f^{-1}Q\), where \(Q=I\setminus f[O\xi]\), is open-and-closed in \(X\). The set \(f^{-1}Q\) is open, since \(f^{-1}Q=f^{-1}(a,b)\setminus [O\xi]\), and it is closed, for \(f^{-1}Q=f^{-1}I\setminus O\xi\). But this contradicts the connectedness of \(X\).

2. \(a\in [(L^-\setminus a)\cap f[O\xi]]\), \(a\notin [(L^+\setminus a)\cap f[O\xi]]\). In this case \(P=f^{-1}(L^+\setminus a)\) is open-and-closed in \(X\). It is obvious that \(P\) is open and that \([P]\setminus P\) either consists of the single point \(f^{-1}a\), or is empty. But at \(f^{-1}a\), by virtue of the conditions of this case, there is a neighborhood containing no points of the set \(P\). Thus, the theorem is proved.

The following theorem also holds; I give it without proof.

Theorem 3. Let a condensation \(f:X\to L\) be given, where \(X\) is a connected and locally connected space, and \(L\) is a line. Then \(f\) is a homeomorphism.

Proof of Theorem 1. We shall call a point \(\xi\in E^n\) bad if the image of at least one neighborhood of the point \(f^{-1}\xi\) contains no neighborhood of the point \(\xi\). It is not hard to verify that the set of all bad points \(W\) is closed in \(E^n\). We shall show that \(W\) is nowhere dense in \(E^n\). Suppose the contrary: an open ball \(Q\) consists entirely of bad points, \(Q\subset W\). Let \(\xi\in Q\), \(x=f^{-1}\xi\), and let a neighborhood \(Ox\) have bicompact closure \([Ox]=B\).

* A condensation is a one-to-one continuous mapping.
** A space is peripherally bicompact if it has a base whose elements have bicompact boundaries.

If some point \(\eta \in fB\) were interior, then it could not be bad; hence \(fB\) is nowhere dense in \(Q\). Since \(X\) is finally compact, it follows from what has been proved that \(Q\) can be represented as the sum of a countable number of closed nowhere dense sets, which contradicts the theorem. Thus \(W\) is closed and nowhere dense.

We shall call a spherical neighborhood \(O\xi\) of a point \(\xi \in W\) special if \(O\xi = O_1 \cup O_2\), where \(f^{-1}O_1 = V\) is open-and-closed in \(f^{-1}O\xi\), and \(V\) is a neighborhood of the point \(f^{-1}\xi\) whose closure is bicompact. Note that \(O_1 \cap |O_2| \subseteq W\). The neighborhood \(V\) of the point \(f^{-1}\xi\) will be called conjugate to \(O\xi\). Every point \(\xi \in W\) has special neighborhoods of arbitrarily small radius. Construct a neighborhood \(OW\) of the set \(W\): mark an arbitrary \(\xi_0 \in W\) with its special neighborhood \(O\xi_0\); let the point \(\eta \in O\xi_0\), \(\eta \notin W\), but \(f^{-1}\eta \in V_0\), where \(V_0\) is conjugate to \(O\xi_0\). The neighborhood \(OW\) is an arbitrary neighborhood of \(W\) that is the sum of special neighborhoods of points of \(W\), each of which, except \(O\xi_0\), does not contain \(\eta\).

Since \(X \setminus f^{-1}W\) is mapped homeomorphically by \(f\) onto \(E^n \setminus W\), the boundary of \(f^{-1}OW\) is mapped homeomorphically onto the boundary of \(OW\). Mark the \(U\)-component of the set \(OW\) containing \(\xi_0\). We note that \(f^{-1}U\) is disconnected, \(f^{-1}U = \Gamma_1 \cup \Gamma_2\), where \(\Gamma_1\) is the sum of the neighborhoods \(V_\alpha\) conjugate to the special neighborhoods forming \(U\), and \(\Gamma_2\) contains the point \(\eta\). If \(R = W \cap U\) does not split \(U\), then \(R\) does not split \(E^n\) either, and then \(f^{-1}R\) is open-and-closed in \(X\), for \(U \setminus R\) is connected, while \(f^{-1}U\) is disconnected. But this contradicts the connectedness of \(X\). Suppose now that \(R\) does split \(U\); then \(R\) splits \(E^n\), since \(U\) was a connected neighborhood of the closed set \(R\), and such a splitting can be carried out:
\[ E^n \setminus R = \Delta_1 \cup \Delta_2, \]
where \(\Delta_1\) and \(\Delta_2\) are open in \(E^n \setminus R\), \(\Delta_1 \cap \Delta_2 = \Lambda\), and also \(f\Gamma_1 \setminus R \subseteq \Delta_1\), \(f\Gamma_2 \subseteq \Delta_2\). It is now not hard to see that \(f^{-1}\Delta_2\) is open-and-closed in \(X\). This contradicts the connectedness of \(X\), and therefore the set of bad points \(W\) is empty. The theorem is proved.

Corollary 1. Every condensation of \(E^n\) onto itself is a homeomorphism.

Corollary 2. Let a condensation be given \(f: X \to I^n\), where \(X\) is connected, locally bicompact, and paracompact, and \(I^n\) is the \(n\)-dimensional cube. Then \(f\) is a homeomorphism.

Corollary 3. Let a condensation be given \(f: X \to S_n\) \((n \ne 1)\), where \(X\) is connected, locally bicompact, and paracompact, and \(S_n\) is the \(n\)-dimensional sphere. Then \(f\) is a homeomorphism.

Corollary 1 follows immediately from Theorem 1, but Corollaries 2 and 3 require proofs, which I omit.

Received
8 III 1963

Note added in proof. Recently I obtained a result more general than Theorem 1:

Theorem. Let a condensation be given \(f: X \to Y\), where \(X\) is a connected, locally bicompact space with a countable base, and \(Y\) is a locally connected, locally bicompact, and unicoherent space with a countable base. Then \(f\) is a homeomorphism.

REFERENCES

1. E. G. Sklyarenko, DAN, 120, No. 6, 1200 (1958).