PROVING COMPLETENESS OF THE HAUSDORFF INDUCED
I would like to acknowledge Professor Russ Gordon and Professor Pat Keef for
their advice and guidance on this project. I would also like to thank Nat
May 5, 7:00 pm
Caution: You must show all works. Otherwise you will not receive a full credit. Without
proofs, you can use any theorems in the textbook and any theorems proved in class. When
you want to use an exe
December 15, 2016
1. Determine the components and path components of R .
pf. Let ~x = (xi )iN , ~y = (yi )iN be elements of R . Then : [0, 1] R
(t) := (1 t)~x + t~y
is a continuous map since each component function is c
Tietze Extension Theorem
Proof of Tietze Extension Theorem
Lecture : Tietze Extension Theorem
Dr. Sanjay Mishra
Department of Mathematics
Lovely Professional University
November 26, 2014
Tietze Extension T
Math 421 2011 Fall
Mldterm Exam November 4
Caution: You must show all works. Otherwise you may not receive full eredttsv Without
proofs, you can use any theorems m the tertbook and (my theorems proved in class If you
want to use an erer
Homework 4 Solutions
Proposition. If cfw_T is a family of topologies on X, then
T is a topology on X.
Proof. Since and X are in each T , they must also be elements of
then C T for all ; it follows that
is a finite subcollection of
December 8, 2016
1. Show that a compact metric space X is 2nd countable.
pf. For given q Q+ , there exists finite number of open balls of radius
q which covers X. Denote Bq by the set of the previous balls. Set B :=
qQ+ Bq . The c
Math 421 F. l E 2011 Fall
General Topology Ina xam December 21
Caution: You must show all works. Otherwise you may not receive full credits. Without
proofs, you can use any theorems in the textbook and any theorems proved in class. If you
want to use an e
November 30, 2016
2. Suppose A1 , A2 , . . . are connected subspaces of a space X such that
An An+1 6= for each n. Show that X = An is connected.
pf. Suppose X = C D is a separation of X. We can assume that A1 is
contained in C by
December 8, 2016
3. Suppose (X, d) is a metric space. If f : X X satisfies d(x, y) =
d(f (x), f (y) for all x, y X, then f is called an isometry.
a. Show that an isometry is injective and continuous.
b. Show that an isometry on a
October 20, 2016
3. Suppose cfw_X1 , X2 , . . . is a sequence in the
Qproduct space X .
a. Show that the sequence converges to X X iff the sequence cfw_ (X1 ), (X2 ), . . .
converges to (X) for each index .
b. Is the statement i
October 13, 2016
1. Suppose X and Y are spaces and y0 Y . Let c : X Y be the constant
function defined by c(x) = y0 . Show that c is continuous.
pf . For any open subset V in Y ,
X, if V contains y0
c (V ) =
, so c
October 12, 2016
Suppose cfw_T is a collection of topologies T on a fixed set X.
a. Show that T is a topology on X.
b. Is T a topoogy? Prove or disprove by a counterexample.
a. Clearly , X T . We have to show that
October 18, 2016
3. Prove the following.
a. A B = A B.
b. A A .
lemma. If A B, then A B. (It is clear since any closed set(e.g. B)
containing B also contains A)
b. Suppose x A , then x A for some . Since A A ,
A A by the abov
Q. Let X = cfw_(x, y) R2 | x2 + y 2 1 and Y = cfw_(x, y) R2 | x2 + y 2
1, x 0 be subspaces of R2 . Show that X and Y are homeomorphic.
pf . Define functions f : X Y, g : Y X by
x + 1 y2
f (x, y) := (
, y), g(x, y) := (2x 1 y 2 , y).
Then g f = IdX