’
62
H．KAWASHIMA
We can obtain it by inspecting the sign of the value
・（・…／）−R（…C・’）一・・劉為＋（書）馴高，一κ歪i隣’・
where Aγ一γi一γ、， AC−cjt−C、’，
晋一∨㌫，［・−ll・2・r2・φ・・（∋一・−ll・2・r21φ・・（・）コ・
器一一1＋・・［・（a−・）一・（b−・）コ＋・・［・（a＋・）一・（b＋・）］
and
砦ξ一θ叢γ2）［ヰ需＋1）φ1・ω＋司＋θ一濃γ”［（鵠＋1）φ2ノ（・）＋M（b）］・
ON THE TOPOLOGICAL PRODUCT OF MINIMAL
且AUSDORFF SPACES
BY
HIRosHI KAWASHIMA
Atopological space（X， T）is said to be minimal、協μ∫∂07ガifτis Hausdorff and
there exists no Hausdorff topology on X strictly weaker than T．
Any compact Hausdor任space is necessarily minimal Hausdor仔， and the converse
is not true． Whose counter example is the space given by Urysohn［1］．
Many characterizations of minimal Hausdor任spaces are given in［1］，［2］etc．
M．P． Berri has proposed the fbllowing problem in［1］．
Is the topological product of minimal Hausdor什spaces necessarily minimal
Hausdorff？
The purpose of this paper is to answer this problem aMrmatively．
The following theorem is one of the characterizations of minimal Hausdorff spaces
［2］．
THEoREM A． Anecessasy and su伍c輌ent condition that a Hausdorff space（．Y， T）
be minimal Hausdor『is that（x，T）is quasi・compact and quasi−regular．
Where， a topological space（X，τ）輌s said to be quaぷi−compact（absolutely closed）
if any open covering of．Y has finite sub−family whose closures fbrms also a covering
ofX．
And， a topological space is said to be quasi−regular if the family of regularly open
sets fbrms a base f（）r the open sets， where a subset A is regula’かoρen if and only
if A coincides with the interior of its closure， that is A−．4−o．
The following theorem is well−known［2］．
ON THE TOPOLOGICAL PRODUCT OF MINIMAL HAUSDORFF SPACES
63
THEoREM B． The topological product of quasi−compact spaces is also quasi−
compact．
Therefore， to answer aMrmatively the above problem it is suHicient to prove the
theor㎝below．
THEoREM C． The topological product of quasi−regular spaces is also quasi−regular．
To prove this theorem， we use next two lemmas． The first of these is well−known．
Let｛Xa｝A∈n be a family of topological spaces and X be their product with usual
product topology．
LEMMA 1． The product of closures of subsets is the closure of the product of the
subsets， that is，π！4λ一＝（llAλ）一．
LEMMA 2． The product of interiors of subsets contains the interior of the product
of the subsets． And if the product of interiors of subsets is not empty and the
subsets coincide with whole spaces except for finite numbers of indexes． Then the
product of interiors of subsets coincides with the interiors of the product of the
subsets， that is， llAlo＝（π．4λ）o when llAaO十・φand Aλ＝XA except fbr finite numbers
ofA．
The Proof of Lemma 2． The first part is obviours． We shall prove theぽond
part． Let x＝（xA）be any point of llAzo． Then xa∈！lao for anyλ， and there exists
aneighbourhoodし㌦of xa wnich is contained in、4a． From our assumption， we
may assume that UA is the whole space為except fbr丘nite numbers ofλ． Then，
17こな is a neighbourhood of x＝（xλ） in the product topology and is contained 量n
π．4λ．
Therefore， x∈（ル4A）o． The proof is completed．
The Proof of Theorem C． Let XA be a quasi・regular space f（）r anyλ． We shall
prove that the product llXA is also quasi−regular．
Let x＝（XA）be any point in llXA andπ｛ろbe a neighbourhood of x． Then， Ua
is a neighbourhood of xλ， and there exists a regularly open set Aλbetween xA and
Ux because of the quasi−regurality of為． Where， We may assume that AA is the
whole space Xa except fbr丘nite numbers ofλ， fbr｛Ua｝has the same property．
Now， according to Lemmas 1，2， we may calculate：
（llAA）一゜＝（πAx−）°＝ZTAλ一゜＝llAa．
Then rLAλis a regularly open neighbourhood of x which is contained in πτな．
Thus UXx is quasi−regular．
Combining Theorem B， C with Theorem A， we obtain the following theorem，
which is the aMrmative answer fbr the problem proposed by Berri．
THEoREM． The topological product of minimal Hausdorff spaces is also minimal
Hausdorff．
輌
H．HIRAKAWA
REIIERENCES
［1］M．P． Berri：Minimal tOpOIOgical spaces， Trans． Amer． Math． Soc．， Vol．108（1963）．
12］N．Bourbaki：TopOlogie G6n6rale， Actualit6s Sci． Ind．， nos．85紐42， Hermann．
ON THE EXISTENCE OF A PERIODIC SOLUTION
・畷＋扁←x・
BY
HuMIKo HIRAKAWA
砦｜
Tllcory of perturbation method on no且・Hロear differential equations was developed
．by Friedrichs． His theory applied on the 2nd order equations is as fb皿ows． 1．et
弐十x＝μ∫（らx，x）be the equation for non−linear osci皿ation， where∫is analytic with
resp㏄t to t， x， Z and periodic wi血resp㏄t to’of period 2π．
He has shown that if
P。（c、。，c2。）＝0，
Q。（e、。，c2。）＝0，
31（：tfe，，裂キo
＾where
P・（・・・・…）−1：7（’・・・・…’＋…S…一・・・・…＋…C・・’）…’〃・
・・（・・・・…）−1㌢（・・・・…’＋…S・叫一…S・・’＋・・・…’）…脇・
．then there exists a periodic solution．
。、蕊、、馳＝二゜1＝蕊還゜㌫麟鰐
introducing tlle new variableμ夕＝＝x−￠10 cos’−c20 sin t， one may replace the give皿
・equation by tlle equivalent one
y＋γ一・f（らc、。COS’＋c2。 Sinら一c、。 Sin’＋c2。00S’）＋μF（’，γ， y，μ）．
Let
・・（・…μ）一！i”F（…γ・・）・i・・tdt
α（a・・bl pt）辛（ら・・y・・）…tdt
K（・…μ）一÷（；81・’（a・・bl pt）−k／°，e’（a・… pt））・
