A modification of the Michael line
by
Akihiro Okuyama
Kobe Women's Junior College, Kobe, Japan

Let (X, \tau) be a Hausdorff space with the topology \tau and Y a subset of X. Then the family {G \cup B : G in \tau, B subset X\Y} defines a new topology on X and we denote it by \tau(X;Y). The Michael line is denoted by (R, \tau(R, Q)), where R is the real line and Q is the subset of R consisting of all rational numbers and \tau is the usual topology on R.

The Michawl line is a very famous and an elegant example whose product with the metric space need not be normal; that is,

The product space (R, \tau(R, Q))×P is not normal, where P is the subspace of R consisting of all irrational numbers.

Now, we modify the Michael line, as below:

Let (X, \tau) be a Hausdorff space and Y a subset of X. Then the family

{G \cup B:G in \tau, B \cap Y=\emptyset} \cup {H:H \text is a G\delta-set in (X, \tau) with Y subset H}

defines a new topology on X, and we denote it by \tau\delta(X, Y).

The space (X, \tau\delta(X, Y)) has a similar properties with (X, \tau(X, Y)) and, on the other hand, it has a very different feature.

From now on we assume that (X, \tau) is always an uncoutable separable metric space and Y is a subset of X. Then we have the following theorems.

Theorem 1. (X, \tau\delta( X, Y)) is a paracompact space.

Theorem 2. If Y is not a G_\delta-set, then (X, \tau\delta(X, Y)) is not a sequential space.

Theorem 3. The product space (X, \tau\delta(X, Y))×Z is normal for any analytic space Z. Especially, (R, \tau\delta(R, Q)) ×P is normal.

Date received: June 3, 2001