Fundations

Topological space (X,T)
x in (X,T) are points

RELATIVE OR INDUCED TOPOLOGYSub topological space of subsets of (X,T)

THE TRACE
system of subsets of a subset on (X,T)

The product topology
Result of create Opens that unite two topologies:

ej. R2= RxR

Density in (X,T)
A dense in X if \A=X

A nowhere dense
if is no dense in any subset

Connected Topological Space
if only Vacio and X are unique
open close subset

FINAL TOPOLOGY
Minimal topology on X that
make any mappinf
fi : X --> Y

Open Sets (O)
Elements of (X,T)

Closed Set
Complements of (O)

properties

Morgan Laws

Neighborhood of
points x in (X,T)

Contact Points
in the Boundary
adherent point
"closure point "
point of closure

Closure of barra A
Set of all contact points of A

Limit Points
not in Boundary

Interior
Larger open set in a set A in (X,T)

Comparison of Topologies

T2 Weaker Topology than T1
if every open of T2
is an open of T1

T1 is Stronger Topology than T2
if every open of T2 is an open of T1

( f ) Mapping between
topological spaced
(X,Tx) to (Y,Ty)

CONTINUOUS MAPPING
if the image of every OPEN W in Y is an
OPEN in X ; this is inv(f(W)) is in Tx

THEROREM : f:(X,Tx) ---> (Y, Ty)
IS CONTINUOUS ALSO IF
Tx is stronger topology than Ty

THEROREM : f:(X,Tx) ---> (Y, Ty)
IS CONTINUOUS ALSO IF
if the image of every CLOSED W in Y is an
CLOSED in X

THEOREM: f: X --> Y, and g:Y --> Z
f continuous map, g continuous map then
g · f : X --> g(f(X)) in Z
IS CONTINUOUS

( f ) Continuous "at point x0"
if yo=f(x0) and for a neighborhood of yo Ny
exist a neigtbohood of x0 Nx such that
f(Nx) included in f(Ny)

Conbtinuous "on X"
f:(X,Tx) ---> (Y, Ty)
if is continuous for all points of X

Continuous Real Mapping
f:(X,Tx) ---> (R,TR)

sets

Real numbers

Ingteger numbers

vertices

(X) Set / Collections

(T) Topology
Family of subsets of (X) Axiomatic definition of open sets (O)

Or alternatively by a axiomatic system of neightborhoods

Trivial Topology (X,T_t)
{X , empty set}

Usual Topology on R
O ∈ T iff ∀x ∈ O ∃ǫ > 0 : ]x − ǫ, x + ǫ[⊂ O.

Discrete Topology T_d
All subset of X
Power set

Intersection of topology is a topology

CONVERGENCY IN (X,Tx)
of sequences of elements xn in X

SEQUENCE xn in X CONVERGES to L if:

For an U in the veninity of L V(L) or open O s.t L in O
enxist a natural numner N S.T.
for all n >= N then xn is in U

xn can converge to several elements of X

For X stronger xn is hard to converge

if Tx is discrete topology
only xn becoming in constant converges

For (X,Tx) to (Y,Ty)
f continuous mapping
xn sequence converges to L in X
if yn=f(xn) then
yn converges to(f(L) in Y

complexes

simplicial

cubic

cell

Homeomorphism (homeomorphic mapping)
is a mapping f: (X,Tx) --> (Y, Ty) s.t.
f is continuous and inv(f) is continuous

(X,Tx), (Y,Ty) HOMEOMORPHIC ≈ Spaces
if there is a homeomorphism between the spaces

Reflexive
(X,Tx) ≈ (X,Tx)

Symetric
(X,Tx) ≈ (Y,Ty) then (Y,Ty) ≈ (X,Tx)

Symetric
(X,Tx) ≈ (Y,Ty) and (Y,Ty) ≈ (Z,Tz)
then (X,Tx) ≈ (Z,Tz)