9.1 Denition. A topological space X satises the axiom T1 if for every points X such
that = there exist open sets U V X such that U, V and V , U.
9.2 Example. If X is a space with the antidiscrete topology and X
3.1 Denition. Let 1 and 2 be two metrics on the same set X . We say that the metrics 1
and 2 are equivalent if for every X and for every > 0 there exist 1 2 > 0 such that
B1 ( 1 ) B2 ( ) and B2 ( 2 ) B1 ( ).
B1 ( 1 )
8.1 Denition. A space X is path connected if for every 0 1 X there is a continuous
function : [0 1] X such that (0) = 0 and (1) = 1 .
The function is called a path in X .
8.2 Example. For any 1 the space R is
1.1 Notation. Sets: A B C
? = the empty set (contains no elements)
N = cfw_0 1 2 the set of natural numbers
Z = cfw_ 2 1 0 1 2 the set of integers
Q = the set of rational numbers
R = the set of real numbers
2.1. Recall that a function : R R is continuous at 0 R if for each > 0 there exists
> 0 such that if |0 | < then |(0 ) ()| < :
A function is continuous if it is continuous at every point 0 R.
5.1 Denition. Let X be a topological space. A set A X is a closed set if the set X r A is
5.2 Example. A closed interval [ ] R is a closed set since the set
is open in R.
R r [ ] = ( ) ( +)
5.3 Example. Let TZ be the Za
Recall: Let X , Y be topological spaces. A function : X U is continuous i for every open
set U Y the set 1 (U) X is open.
6.1 Proposition. Let X , Y be topological spaces. A function : X Y is continuous i for
every closed set
7.1. Let [ ] R be a closed interval and let ( ) R be an open interval. We would like
to verify that [ ] and ( ) are non-homeomorphic topological spaces. The idea of a proof
of this fact is as follows. Assume that there exists
12.1 Denition. Let X be a topological space.
1) A cover of X is a collection Y = cfw_Y I of subsets of X such that
Y = X :
2) If the set Y is open in X for all I then Y is an open cover of X .
3) If a co
We have seen already that a product of nitely many compact spaces is compact (13.6) . The
main goal here is to show that the same is true for arbitrary products of compact spaces:
15.1 Tychono Theorem. If X is a compact space
We have seen that compact Hausdor spaces have several interesting properties. For this
reason this class of spaces is of special importance in topology. If we are working with a
space X that is not compact then we can still a
10.1 Urysohn Lemma. Let X be a normal space and let A B X be closed sets such that
A B = ?. There exists a continuous function : X [0 1] such that A 1 (cfw_0) and
B 1 (cfw_1).
10.2 Lemma. Let X be a normal space, let A X be a
The Heine-Borel Theorem 13.3 describes compact subspaces of R . We have previously
noticed (13.4) that this theorem is not true if we replace R by an arbitrary metric space.
Our next goal is to give a description of compact s
Recall that a topological space X is metrizable if there exists a metric on X such that the
topology on X is induced by . Our goal here is to show that if a topological space satises
a couple of conditions then it is metrizab
We have seen already that a closed interval [ ] R is a compact space (12.11). Our next
goal is to prove Heine-Borell Theorem 13.3 which gives a simple description of all compact
subspaces of R .
13.1 Denition. Let (X
4.1 Denition. Let X be a set. A basis on X is a collection of subsets of X satisfying the
1) X = V B V ;
2) for any V1 V2 B and V1 V2 there exists W B such that W and
W V1 V2 .