TÃÂpàC NhÃÂn TÃÂi NÃÂng àThi Cu I H C Kì
TÔPÔ-C NHÂN TÀI NĂNG
H c kì II, 2007–2008
Giáo viên: Huỳnh Quang Vũ
Đ Thi Cu i H c Kì
Th i gian: 180 phút. Đư c s d ng bài gi ng c a giáo viên và bài ghi c a sinh viên.
Không đư c s d ng các tài li u khác và máy vi tính.
1. Let F = R or F = C.
A polynomial in n variables on F is a function from Fn to F which is a finite
sum of terms of the form axm1 xm2 · · · xmn
1
2
n , where a, xi ∈ F and mi ∈ N. Let P be
the set of all polynomials in n variables on F.
If S ⊂ P then define Z(S) to be the set of common zeros of all polynomials in
S, thus Z(S) = {x ∈ Fn/ ∀p ∈ S, p(x) = 0}. Such a set is called an algebraic set.
(a) Show that if we define that a set in Fn is closed if it is algebraic, then this
gives a topology on Fn, called the Zariski topology.
(b) Show that the Zariski topology on F is exactly the finite complement
topology.
(c) Show that if both F and Fn have the Zariski topology then all polynomials
on Fn are continuous.
(d) Is the Zariski topology on Fn the product topology?
2. (a) Fix a point x = (xi) ∈ ∏i∈I Xi. Define the inclusion map f : Xi → ∏i∈I Xi
by
xj
if j = i
y → ( f (y))j =
y
if j = i
The inclusion f is a homeomorphism to its image (an embedding of Xi).
The image of the inclusion map is ˜
Xi = ∏j∈I Aj where Aj = {xj} if j = i and
Ai = Xi. Thus ˜
Xi is a “parallel” copy of Xi in ∏i∈I Xi.
The spaces ˜
Xi have x as common point.
Use this result to prove the followings.
(b) If ∏i∈I Xi is Hausdorff then each Xi is Hausdorff.
(c) If X and Y are connected then X × Y is connected.
3. We could have noticed that the notion of local compactness as we have de-
fined is not apparently a local property. For a property to be local, every neigh-
borhood of any point must contain a neighborhood of that point with the given
property (as in the cases of connectedness and path-connectedness).
Show that for Hausdorff spaces local compactness is indeed a local property.
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