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Department of Mathematical Sciences
Mellon College of Science
1967
Natural covers
S. P. Franklin
Carnegie Mellon University
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NATURAL COVERS
by
S. P. Franklin
67-29
September, 1967
NATURAL COVERS
by
S. P. Franklin
§0.
INTRODUCTION.
In recent years the category of
has come to occupy a fairly central place in topology
k-spaces
(see
for example Whitehead [20], Gale [15], Michael [18], Cohen [9],
Morita [19], Arhangel1skii [l], Bagley and Yang [5], Duda [10],
Whyburn [21], Arhangel1skii and Franklin [3] and many others).
R. Brown suggests this category may serve all the major purposes of topology [8]. Of more recent vintage is the interest
in sequential spaces (see for example Kisyn'ski [17], Dudley
[11],
Aull [4], Franklin [I3],[l4], Boehme [7], Baron [6],
Arhangel1skii and Franklin [3], Fleischer and Franklin [12]).
Several people have noted (and exploited) the similarity of
the theorems which can be proved about
spaces.
k~spaces and sequential
In this paper we offer a general theory which en-
compasses the common aspects of the theories of
and sequential spaces and of others also.
k-spaces
The ideas involved
can be traced from Michael1s generating collections [18],
thru Cohen's weak topologies [9] and Brown's natural covers [8].
We strengthen Brown's concept slightly, retaining his terminology.
Section 1 is devoted to the statements of some preliminary
facts and lemmas due either to Brown [8] or to Cohen [9] .
The theory is then developed in sections 2 and 3.
HUNT LIBRARY
CARNEGIE-MELLON UNIVERSITY
§1.
THE WEAK TOPOLOGY OF A COVER.
Let
topology
E
be a cover for a topological space
T .
The family
which intersect each
S e £
in an
with the relative topology from
X
finer than
X
S-open set (i.e. open in
T )
The restriction of
the same topology on
with
E ( T ) of those subsets of
S
T .
X
is a topology for
£ ( T ) to
S
yields
S (i.e., E(T)/S = T / S ) , and repetition
of the process leads to no new open sets
(i.e.,
£(£(T))
= S(r))
One key fact is given in the following.
1.1
LEMMA.
A function out of
X
only if^ its restriction to each
If we let
the spaces
0
£
E(r)~continuous if and
S e £
ijs continuous.
denote the disjoint topological sum of
S e £ , the inclusion maps
yield a quotient map
for a space
Xf
<p: ^
E—>.X .
with topology
T*
is the associated quotient map.
is a cover of
1.2
is
X x Xf
LEMMA^ with
<p X <p» : i'+} E X Q
S c x
Suppose
and that
Then
combine to
S!
is a cover
<p! :
©EI~>XI
. X X £' = (S X S'|S e S, S
and
X X Xf
topologized by
S1 — > X X X '
(S X ZJ1 ) (T X T ! ) ,
is a quotient map.
Further we have
1.3
LEMMA.
E ( T ) x S' (T» ) c (E x Ef ) (r x r) .
is^ a^ £(r)~interior point of some
each
if: each
x e X
S e E ^ and similarly for
xf e Xf , then equality holds.
The next lemma will prove central to our theory as it
allows the construction of a coreflexive functor.
1.4
KEY LEMMA.
rf for each
Sf e £f
i) there is an
is continuous,, then
and
S e £ , f : X -> Y
f(S) 5 ST
with
f
and
L 5 X1
±s^ continuous with respect to £ ( T )
or if
£
£
and
£'
are covers of
is a refinement of
If
A c X , let
LEMMA.
^
^^^
£(T)
|A <==
If
Ef (r ) c £ ( T ) .
£ ( T ) = £T (r) .
£|A = {A 0 s|S e £} , and let
be the relative topology on
1.5
X .
£? , then
Hence if each is a refinement of the other
G
ii) f |S:S~>S'
£'(T>).
Now suppose both
S
satisfies
T|A
A .
(£|A)(T|A)
. Further, if. A
T-clbs'edj then equality holds and
A
and each
i^ E ( T )
closed.
Refering to Example 5.1 of [13] , let
E
be the collection
of all convergent sequences (with limit points) in
A = M\N
.
Each
€ (£|A)(r|A)
S e £
is closed but
but not to
and let
is not, and
{0}
£ ( T ) |A .
For another example^ let
[0,1]
A
M
I
be the closed unit interval
with the usual topology, and let
of all connected subsets of
be the collection
Let
A
be the Cantor set .
A , being compact, is closed but each
S
need not be.
I
G
£ , T = £(r) . But
is discrete.
§2.
r|A
I .
S
is compact while
(£|A)(T|A)
Thus the conditions of lemma 1.5 are needed.
NATURAL COVERS AND THEIR SPACES.
By a natural cover we shall mean a function
assigns to each topological space
1)
Since
if
S e 2L
and
S
X
a cover
£
IL
is homeomorphic to a subset
which
satisfying
T
to
Y ,
-then
T e L
S € £
, and
2)
there is a
may choose
£v
if
f: X ~ * Y
T G L
with
is continuous and
f (S) 5
T . For example we
to be the compact subsets of
X , or the
connected subsets, or the countable subsets, or the convergent
sequences.
The first and last of these are the notivating
examples which lead to the
Now to each space
the space
k~spaces and the sequential spaces
X
with topology
T
we may associate
cxx , the same set of points topologized by
We may also assign to each continuous function
the continuous (by Lemma 1.4) function
Since
0
of = f , a
f : X—~>Y ,
f = erf : crx—^aY. .
preserves compositions and identies, i.e.,
1*L 1L f^nctor from the category
\Tj of topological spaces
into itself.
Let us call a space
(i.e.
T = E (T)) .
If
X
L
a
S~space whenever
is the natural cover which assigns
to each space its compact subsets, the
the
k-spaces.
If
E
aX = X
S-spaces are precisely
assigns the convergent sequences, the
£-spaces are the sequential spaces.
(This is almost immediate.)
2.1
jls a, 2>space
LEMMA.
Hence
a
For each space
X , aX
is^ a, retraction from
^
(acrX = aX) .
onto the category
S-spaces.
Proof.
Clearly
£X(T) ESaX(£X(r))
is a subspace of
o*X
^x E Eax •
E
Hence
*
Now
and hence belongs to
a x ( E x ( r ) ) E hShS7^
if
S G T
*X >
S
= E
v
, i.e.,
x(r)
and t h e
lemma is proved.
2.2
LEMMA.
S
(g) i£ a. coreflexive subcategory of
T) .
Proof.
lv : X — > X
and
is a
Y
is continuous.
A
f \ Y—>aX
1Y .
2.3
f : Y ~~^>X
is continuous
S-space the following diagram commutes.
^ ->X
Y-
Hence
If
r
Y
r
T^^T^
*
is continuous and uniquely factors
f
thru
This completes the proof.
PROPOSITION.
The category
(§} o£
S~spaces i£ closed
under quotients and disjoint topological sums.
Proof.
This is a direct consequence of Lemma 2.3 and Theorem
A of Kennison [16] since the disjoint topological sum is the
coproduct in @
.
It is not at all difficult to give a direct
proof.
2.4
a
2.5
COROLLARY.
Every open or closed image of _a &-space is
TJ- space*
COROLLARY.
When a, product space i^s a^ £-space,, so
is each of its factors.
2.6
COROLLARY.
The continuous image of a compact
in a Hausdorff space is ^
2.7
COROLLARY.
S-space
2>space.
The inductive limit o«f any system o£
I^spaces
is again ji 2>space.
2.8
COROLLARY.
Any adjunction space of
S~spaces is a
Of course Proposition 2.3 and its corollaries may be
immediately asserted for
k-spaces and sequential spaces.
E-space.
2.9
LEMMA.
For each
X , ©
E
is, a, E-space.
X
Proof.
If
fore
S e L
S = OS .
, then
S 5 S
Hence by 2.3
implies
±> £ x
is a
This allows a characterization of
2.10
PROPOSITION.
natural mapping
Proof.
X
1+) IL
quotients, if
cp
and there-
E-space.
E-spaces as follows.
isi a. E-space if and only if the
^ x ^x
<P : ®
Since
S e Sg
is a
—
fL quotient mapping.
S-space and
is a quotient map,
X
S
is closed under
is a
E-space.
The converse has already been noted.
Hence we get the sequential spaces as quotients of
zero-dimensional, locally compact metric spaces and the kspaces as quotients of locally compact spaces.
(Lemma's 1.2 and 1.3 together with Proposition 2.10
provide a criterion for the product of two
be a
E-space.
E-spaces to again
Although possibly useful in particular cases,
it is clumsy to state and we shall omit it.)
This characterization is in fact more useful than it
would appear.
For example, with it we can say something
about subspaces of
22-spaces..
2.11
If
^
€
PROPOSITION.
^Y
X
and every open subset of each
ifL IL E~space, then every open subset of
X
is a
E-s-pace.
Proof.
Suppose
U
the natural mapping.
open in
<P~ (U)
is open in
X
and
Then for each
S
and hence is a
is a
E-space and
E-space.
<p : gi E y ^ X
S e L
, <p
is
(U) H s
is
Thus their topological sum
^ = <p|<p~ (U)
maps
(p^CU)
onto
U . By 2.10 it is enough to show
For
V 5 U , if
^ rl (V)
^
to be a quotient map.
is open in ^""1(U) , then for each
S e £ v , ^ rl (V) 0 S - ^"1(V) (1 (p^OJ) n S
is open in
A
Since
X
is a
ID-space,
V
is open in
X
and hence
S .
U , and
we are done.
From 2.11 one sees at once that open subsets of sequential
spaces and
k-spaces are again sequential spaces and
It is also true that being a sequential space or a
is a local property.
2.12
PROPOSITION.
Let
Ij[ each
X
and
A e £ Y (r)
x e X
has an open neighborhood
jLs a. S~space.
choose
x
A
open
e A .
x. . If
S e L
O
is open in
union of the
U
f|s
f
is continuous.
The
2.13
L
X . But
, then
A
3
r \ S - (T |U) | S
A
is the
f : X—>Y
%~continuous just in
S G L
.
It follows from
±s^ S-continuous if and only if f : crx—>Y
£-spaces can be characterized in terms
£-continuous functions.
PROPOSITION.
X
is; a. £~space ije and only if every
^-continuous function out of
Proof.
c
S <= U e T
and hence in
is continuous for each
Lemma 1.1 that
of the
be an
A 0 U , and we are done.
Let us call a function
case
U
U ~-~
S fl (n 0 A) = Sfi A € T | S . But since
A fi U
Let
O
£-space neighborhood of
Thus
k-space
This follows from
which is _a S-space, then
Proof.
k-spaces.
If each
l x : X —~>aX
Conversely
X
^continuous function is continuous, then
is continuous and
f
is continuous.
Is continuous.
being
But
X
is homeomorphic to
£-continuous implies that
X = aX
since
X
is a
f :
a-space.
aX .
8
Following 3 we may assign an ordinal number to each
in a topologically invariant manner.
A
a subset of
X .
Define
A° = A;Att = (A^>
let
wise.
The
A
Let
X
be any space and
= U(ci(A n S)|S e S Y } .
a = j8 + 1; A a = U{AP|jS < a]
if
S-space
S-characteristic of
Now
other-
X
is the least ordinal a (if
a
it exists) such that for each subset A of X , A = c ^ x ^ '
(See [3] for the existance of sequential and
Again we may characterize the
2.14
PROPOSITION.
X
i_s a
k
characteristics.)
S-spaces.
Z-space JLf and only if_ jU has a_
^" characteristic.
Proof.
If
X
is a
£~space, it suffices to show that for each
OL
A ^ X
there is an ordinal
a
such that
simply takes the sup of such ordinals.)
A
= cl^A .
(One
Clearly for each
OL , A c: A ^ clYpA . If A
is not closed, there is some
S e ZL with A H s not closed in S , i.e. there is some
point in
AP
\ AP
.
Hence by cardinality, some
closed and hence equals
clYA .
Conversely if
A
X
is
has
£-
A
characteristic
OL and
A fl S
A
then
A
is a
£~space.
^A
and hence
A
For any natural cover
0
is" closed in
S
for each
c A
and
E , a
if and only if it is discrete.
spaces and of
special study.
A
is closed.
S-space has
Thus
^characteristic
< 1
(called
kr-spaces respectively) have received
We formalize the common part of these theories
in the next section.
ST-SPACES AND HEREDITARY QUOTIENT MAPS.
A mapping
X
In the case of sequential
k-spaces, those of characteristic
Frechet spaces and
§3.
S e
OL
f : X—VY
is an hereditary quotient map
if and only if for each subspace
YJ
of
Y , the mapping
9
f
Arhangel1 skii showed that
= f |f~~ (Y.j)is a quotient map.
° P e n maps and closed maps are hereditary quotient
and characterized them as follows:
a mapping
f : X
an hereditary quotient map if and only if for each
for each open neighborhood
point of
X
3.1
f(U)
U
of_ f~ (y) , y
y e Y
and
i^ ail interior
([2]).
£T-space if its
is called a
PROPOSITION.
If
X
Suppose
A c y
and
f""1(y) 0 cl f"1(A) ^ 0 .
L-characteristic is
Sf - space and
isa
an hereditary quotient map, then
Proof.
>Y ±s^
f : X—^Y
< 1 .
i£
ijs JEI Sf-space.
Y
y e cl A . We claim that
(If it were empty,
U = X \ cl f"1(A)
v
-1
would be an open neighborhood of
f
(y)
and by Arhangel 1 skii 1 s
characterization of hereditary quotient maps,
interior point of
hence
f (U) c Y \ A
f(U) .
E
since
c
y e cl A.)
Then for some
Then
f : S—>S T
and
Choose
S eZ^
is a natural cover, there is some
f(S) c s1 .
would be an
X \ cl f"1(A) <~X \f~1(A)
contradicting
x € f^Cy) n cl f"1(A) .
Since
But
y
, x € cl g (S fl f""1(A))
S'e L
with
y e f [cl Q (f 1 (A) n S) ] c cl Qf [f (f"1(A) 0 S) ] ,
is continuous.
Thus
y e cl e . (A flf(S))
clg, (A (1 ST ) and we are done.
The preceeding proposition and the suceeding lemmas are
aimed at characterizing
E'-spaces in terms of hereditary
quotient maps.
3.2
LEMMA.
Every disjoint topological sum <o£ Sf - spaces
is again _a Sf -space.
10
3.3
LEMMA.
Ijf
X € 2^ , then
3.4
LEMMA.
Each
3.5
LEMMA.
For each
3.6
PROPOSITION.
S e ZL
X
i£ a
S'-space.
.is «* £'- space.
L 1 -space.
X , (& IL- is a
i£ a. £f - space if_ and only jLf <P :
X
S
is an hereditary quotient map.
Proof.
Proposition 3.1 and Lemma 3.5 yield one direction
immediately.
For the converse we will again use Arhangel*skii1s
characterization.
U
Suppose
X
is an open neighborhood of
<p~ (x)
x € cl(A<p(U))) „ then for some
Hence^ regarding
contradicting
S
in
U
<p(U)
x e X
<£> 2L- .
5
and
If
S e Z^ , x e cl g (S n(xVp(U))) .
as a subspace of
cp~ (x) ^
interior point of
Lf-space ,
is a
and
U
IB 2L
open.
x e cl(SMJ)
5
Hence
x
is an
and we are done.
Since the Frechet spaces are precisely the hereditary
sequential spaces ([14] Proposition 7.2), one might look for
a similar result for any natural cover.
Unfortunately., using
the connected sets to generate the natural cover,, an example
can be constructed of a
not a
£-space.
S!-space with a subspace which is
However something can be said in the general
case.
3.7
^
PROPOSITION.
If each subspace of
X
isa
£-space, then
iiL -BL S* -space.
Proof.
Let
tp :
y
£
T^~>X
be the natural mapping,
X^
be
1
a s u b s p a c e of
X
and
<p = <p <p~ (X n )
.
Let
<pn : •+»
J.
U <= x and <p " 1 (U) is open in
—
o
1
1
<P ~ (U) = <P~ (U) n ±s ZL.
is open in
be the natural mapping.
V
(X ) , then
1
J.
If
O
A,
11
But
<p
<p-
is a quotient map and so
is a quotient map and
<p
U
is open in
X^ .
Thus
an hereditary quotient map ,
and 3.6 completes the proof.
3.8
COROLLARY.
rf every subspace o£ a. Z)r - space .is _a S-space,,
then every subspace is a
ST - space.
REFERENCES
[l]
A. B. Arhangel1skii, Bicompact sets and the topology
of spaces, Dokl. Akad. Nauk, SSSR, 150(1963), 9.
[2]
A. B. Arhangel1skii, Some types of factor mappings and
the relation between classes of topological spaces,
Soviet Math. Dokl., £(1963), 1726-1729.
[3]
A. B. Arhangel1skii and S. P. Franklin, Ordinal invariants
for topological spaces, to appear.
[4]
C. E. Aull, Sequences in topological spaces, Not. Amer.
Math. S o c , _12(l965), 65T-79.
[5]
R. W. Bagley and J. S. Yang, On k-spaces and function
spaces, Proc. Amer. Math. S o c , 1/7(1966), 703-705.
[6]
S. Baron, The coreflective subcategory of sequential spaces,
to appear.
[7]
T. K. Boehme, Linear s~spaces, Proceedings of the
Symposium on Convergence Structures, University of Oklahoma, 1965.
[8]
R. Brown, Ten topologies for X X Y , Quart. J. Math.,
Oxford, (2) 14/1963), 303-319.
[9]
D. E. Cohen, Spaces with weak topology, Quart. J. Math.,
Oxford, (2) 5/1954), 77-80.
[10]
E. Duda, Reflexive compact mappings, Proc. Amer. Math.
S o c , 17/1966), 688-693.
[ll]
R. M. Dudley, On sequential convergence, Trans. Amer.
Math. S o c , 112(1964), 483-507.
[12]
I. Fleischer and S. P. Franklin, On compactness and
projections, to appear.
[l3]
S. P. Franklin, Spaces in which sequences suffice.
Fund. Math., 57/1965), 107-116.
[14]
S. P. Franklin, Spaces in which sequences suffice II,
Fund. Math., to appear.
[15]
D. Gale, Compact sets of functions and functions rings,
P r o c Amer. Math. S o c , 1/1950), 303-308.
[16]
J. F. Kennison, Reflective functors in general topology
and elsewhere, Trans. Amer. Math. S o c , 118(1965), 303-315.
[17]
J. Kisynski, Convergence du type L , Coll. Math., 7(1959),
205-211.
[•18]
E. Michael, Locally multiplicatively-convex topological
algebras, Mem. Amer. Math. S o c , 1JL(1952) .
[1.9] K-Morita, On decomposition spaces of locally compact
spaces, Proc. Japan. Acad., 32(1956), 544-548.
[20]
J. H. C. Whitehead, Combinatorial homotopy, Bull. Amer.
Math. S o c , 55/1949), 213-245.
[21]
G. T. Whyburn, Directed families of Sets and closedness
of functions, Proc. Nat. Acad. Sci., U.S.A., 54(1965),.
688-692.
CARNEGIE-MELLON UNIVERSITY
PITTSBURGH, PENNSYLVANIA
HUNT LIBRARY
CONEfilMfELLON UNIVERSIh