拓扑学Topology

Topology
{}4. (a) If is a family of topologies on X, show that is a topology on X.
Is a topology on X?
a a a
燎 攘
13. Show that X is Hausdorff if and only if the diagonal = {x x x
X }is closed in X X.
次
00011. Let F : X Y
Z. W e say that F is contin uous in each variable separately if
for each y in Y , the m ap h : X Z defined by h (x) = F(x
y ) is continuous,
and for each x in X , the m ap k : Y Z defined by 串 ®0 k(y) = F(x y) is continuous. Show that if F is continuous, then F is continuous in each variable separately.
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2. Show that x in the dictionary order topology is metrizable.
2. (a) Let p : X Y be a continuous m ap. Show that if there is a continuous m ap f : Y X such that p of cquals the identity m ap of Y , then p is a quotient m ap.(b) If A X , a retraction of X onto A i ®®Ìs a continuous m ap r : X A such that r(a) = a for each a A. Show that a retraction is a quotient m ap.
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113. Let :
be projection on the first coordinate. Let A be the subspace
of consisting of all points x
y for w hich either x 0 or y = 0 (or both);
let q : A be obtained by restricting p p 串创 ® . Show that q is a quotient m ap that is neither open nor closed.
2
4. (a) Define an equivalence relation on the plane X = as follows:
*
2
2
Let X be the corresponding quotient space. It is homeomorphic to a familiar space; what is it? [Hint: Set g(x y) = x + y .]
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(b) Repeat (a) for the equivalence relation
5. Let p : X Y be an open map. Show that if A is open in X , then the map q : A p(A) obtained by restricting p is an open map.
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K
K
K
6. R ecall that denotes the real line in the K -topology. (See $13.) Let Y be
the quotient space obtained from by collapsing the set K to a point; let
p : Y be the quotient m ap.
(a) Show that Y sa ®
I K
K
K
K
tisfies the T axiom , but is not H ausdorff.(b) Show that p p : + Y Y is not a quotient m ap. [H int: T he
diagonal is not closed in Y Y , but its inverse im age is closed in .]
创 创
6. Let A X. Show that if C is a connected subspace of X that intersects both A and X - A, then C intersects Bd A.
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1. (a) Show that no tw o of the spaces (0, 1), (0,1], and [0, 1] are homeomorphic.[H int: W hat happens if you remove a poin t from each of these spaces?)]
4. Show that every com pact subspace of a m etric space is bounded in that m etric
and is closed. Find a m etric space in w h ich not every closed bounded subspace is com pact.
5. Let A and B be disjoint com pact subsp aces of the H ausdorff space X. Show that there exist disjoint open sets U and V containing A and B, respectively.
17. Show that if Y is compact, then the p rojection : X Y
X is a closed map.p 串
3. Let X have a countable basis; let A be an uncountable subset of X. Show that uncountably many points of A are limit points of A.
4. Show that every compact metrizable sp ace X has a countable basis. [H int:Let A, be a finite covering of X by l/n-balls.]
5. (a) Show that every m etrizable space w ith a countable dense subset has a coun table basis.(b) Show that every m etrizable Lindelof space has a countable basis.
5. Let f, g : X Y be continuous; assume that Y is Hausdorff. Show that {x |f (x) = g(x)}is closed in X .
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