topology

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Math 1510

luonut Rebecca Mallory

MAT 156 Mindmap

luonut Yvonne Cabrera

ospf tech

luonut ABHISHEK KUMAR

Shroud

luonut Mike Ton

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

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

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

Reflexive (X,Tx) ≈ (X,Tx)

complexes

cell

cubic

simplicial

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

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

if Tx is discrete topology only xn becoming in constant converges

For X stronger xn is hard to converge

xn can converge to several elements of X

(X) Set / Collections

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

Intersection of topology is a topology

Discrete Topology T_d All subset of X Power set

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

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

Or alternatively by a axiomatic system of neightborhoods

sets

vertices

Ingteger numbers

Subtopic

Real numbers

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

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

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

( 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)

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

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

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

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

Comparison of Topologies

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

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

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

Neighborhood of points x in (X,T)

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

Limit Points not in Boundary

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

Closure of barra A Set of all contact points of A

properties

Morgan Laws

Closed Set Complements of (O)

Fundations

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

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

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

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

A nowhere dense if is no dense in any subset

The product topology Result of create Opens that unite two topologies: ej. R2= RxR

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

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