Download ON THE SEPARATELY OPEN TOPOLOGY 1. Introduction and

ON THE SEPARATELY OPEN TOPOLOGY
ZBIGNIEW PIOTROWSKI, ROBERT W. VALLIN, AND ERIC WINGLER
Abstract. Replace this text with your own abstract.
1. Introduction and Definitions
This paper deals specifically with topologies on the space R × R. Specifically, the
usual Euclidean topology and the separate topology. In this paper we will compare
and contrast the two topologies and the Gδ sets formed by each.
Let f be a function from R × R into R. We say that f is continuous with respect
to x (with respect to y) if the restricted function fy (x) = f (x, y) where y is fixed
(fx (y) = f (x, y) where x is fixed) is a continuous function from R into R. If f is
continuous with respect to both x and y, then f is called a separately continuous
function. The canonical example of a function which is separately continuous at a
point where it is not continuous is
½ 2xy
(x, y) 6= (0, 0)
x2 +y 2
f (x, y) =
.
0
(x, y) = (0, 0)
Since f is not continuous at (0, 0) we know f −1 (−a, a) (where a > 0) is not an
open Euclidean set in the plane, and a natural question is, “What does f −1 (−a, a)
look like?” The answer is a separately open ball about the origin.
Definition 1. The separately open ball about (a, b) of radius ε > 0 is
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Bε+ (a, b) = (x, b) ∈ R2 : |x − a| < ε ∪ (a, y) ∈ R2 : |y − b| < ε .
(Note: We shall use Bε (a, b) to denote a Euclidean open ball about (a, b) .) It is
obvious the Euclidean open sets are separately open. The following example shows
that the converse is not true.
Example 1. The Maltese Cross
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A = {(0, 0)} ∪ (x, y) ∈ R2 : |y| > |3x| ∪ (x, y) ∈ R2 : |y| < |x/3|
is a separately open, but not Euclidean open, set.
The Maltese Cross has only one point where it is not open in the usual sense.
While we cannot change that it does mean we can describe it in terms “related” to
open sets.
Example 2. The Maltese Cross is a Gδ set in the Euclidean topology. If we let
An = A ∪ B1/n (0, 0), then each An is Euclidean open and ∩An = A.
1991 Mathematics Subject Classification. Primary 05C38, 15A15; Secondary 05A15, 15A18.
Key words and phrases. Keyword one, keyword two, keyword three.
Thanks for Author One.
Thanks for Author Two.
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ZBIGNIEW PIOTROWSKI, ROBERT W. VALLIN, AND ERIC WINGLER
We can extend this idea a little bit to where A is not Euclidean open at more
than one point.
Theorem 1. If A is a separately open subset of R2 and is Euclidean open at all
but finitely many points, then A is a Gδ set in the Euclidean topology.
Proof. Let (xk , yk ) represent the points where A is not Euclidean open. Define An
by
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An = A ∪ B1/n (xk , yk ) .
Then each An is Euclidean open and A = ∩An .
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Question: How far can we extend the exceptional set? Countable? Nowhere
dense?
It is not the case, though, that every separately open set is a Euclidean Gδ one.
Example 3. The Young & Young set comes in here somewhere.
Example 4. Let S = {(x, x) : x ∈ R \ Q} and let G = R2 \ S. Then each x-section
and each y-section is open in R so G is separately open. However, G is not a
Euclidean Gδ set because if it were, then G ∩ {(x, x) : x ∈ R} would be a Gδ subset
of the line y = x. This is impossible since this set is homeomorphic to Q.
Corollary 1. If f is separately continuous and the points of discontinuity are
nowhere dense, then f is Baire class one.
Remark 1. Actually, Lebesgue proved that every separately continuous function is
Baire one, but our method will not extend that far.
Remark 2. The set of points of discontinuity being nowhere dense does not in
itself mean that this set is Baire one.
Theorem 2. Let f : A → R be a separately continuous function where A ⊂ R2 .
Then there exists a separately continuous extension of f to a separately Gδ set.
Proof. Let p = (x0 , y0 ) be a point in the closure of A. Define ω + (p), the separate
oscillation of f at p, as the oscillation considered only over the set
((x0 − ε, x0 + ε) × y0 ) ∪ (x0 × (y0 − ε, y0 + ε)) .
Define A∗ , a subset of the closure of A, as the set of points p in the closure with
ω + (p) = 0. For each p ∈ A∗ assign the sequence {pn } with pn → p and pn =
(x0 , yn ) or pn = (xn , y0 ). Now define the set En by
En = {pn , pn+1 , . . .} .
Since ω + (p) = 0 we have
lim (diam (f (En ))) = 0.
n→∞
So {f (pn )} is a Cauchy sequence whose limit we’ll denote as f ∗ (p). Thus f ∗
the extension of f onto A∗ . Separate continuity follows directly from ω + (p)
0.The set of points in A where ω + (p) = 0 is the intersection of the sets An
{p ∈ A : ω + (p) ≤ 1/n}, a separately open set.
is
=
=
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Example 5. Let
½
f (x, y) =
2xy
x2 +y 2
0
(x, y) 6= (0, 0)
.
(x, y) = (0, 0)
Let Q = {r1 , r2 , . . .} be a listing of the rational numbers, and let
∞
[
f (x − rn , y − rn )
g (x, y) =
.
2n
n=1
Then g is separately continuous, but discontinuous at each point of the form (rn , rn ).
In general, the separately open topology is formed as follows:
Qn Let X1 , X2 , . . . , Xn
be a finite collection of topological spaces and let X = i=1 Xi . We say that
S ⊂ X is separately open provided that for each x = {x1 , x2 , . . .Q
, xn } ∈ S and each
n
i = 1, 2, . . . , n there is a neighborhood Ni of xi in Xi such that i=1 Ni ⊂ S where
Nj = {xj } when i 6= j. For more see [1] and [2].
References
[1] J.E. Hart and K. Kunen, On the regularity of the topology of separate continuity, Topology
and its Applications
[2] M. Henriksen and R.G. Woods, Separate versus joint continuity: a tale of four topologies,
Topology and its Applications
Department of Mathematics, Youngstown State University, Youngstown, OH 45044
E-mail address: [email protected]
Department of Mathematics, Slippery Rock University of PA, Slippery Rock, PA
16057
E-mail address: [email protected]
Department of Mathematics, Youngstown State University, Youngstown, OH 45044
E-mail address: [email protected]