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

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13. Show that X is Hausdorff if and only if the diagonal = {x x x

X }is closed in X X.

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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.

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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 .]

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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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