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marital Pointset topology TopologicalManifold Smooth

A dim Manifolds in IR MAR MEIR is a K dim manifold in if

APE M FU neighhdofp and HEIR VER Lapiece DIRK that isopen and a function f U satisfying Thatisdeformed 1 fisc Recall 2 V IS a 2 fisa homeomorphism f 3 Df ish rank a jIIHpf UES tpqpfmxox.FIsetfallnectors starting U

7 DAIRY fCq P Tps PM Image f Rfy s TqV i Dfw 2 g SEIR dim 2 Since Dfisl l the tangentspace is welldefined andis Jthe samedimension TTT Remake We want K dim manifolds in IR to satisfy the more general def that we are trying to come up with

Point Set To p o l o g y

Let M be a set

Need the notionofcontinuity compact neighbourhood connected a

I hit in an a these notions To p o l o g i c a l structure

first Notice i Continuity in IRA could be formulated only using A opensets Let T E PCM satisfying i

il Q M ET 2 T isclosed under unions and finite intersections

If so we say MT is a topological space with topology T

Appendix Reimann Read A mandatory

To p o l o g i c a l Properties

1 closed AEM is closed it A isopen

2 Compact 3 connectedness 4 continuous i f i M M i cont if f A is open whenever A isopen 5 Bounded Not atopology property It depends on the fact Ad PhxRn R Rq tellfqH 6 differentiable function Not a topological property It depends on V S ofIRN We don'thave a notion ofaddition subtraction

Reina A property istopological iff it stays inramat under homeomorphisms topological invariant

homeomorphism A e B

Subspace To p o lo g y

If M T is a topological space then AEM inherits a topology from the ambient space

Characterized by Unique topology s t c t i finish where i A M i pl p let Ta Anu UET

TA is called the subspace topology

exc i Ven b TA is a topology on A

Based

Ded Let MT be a topological space BET is called a basis for T if forevery VET andPEO 3 BED s t PE B E U

equivalently Bisabasis for T if every VET is a union of elements in B Rn EI RM B Bros PE reft topology is a basis for the regular

In fact we can find a countable basis B Berle Pean real Defra Mrt is second countable if it admits a countable basis

Proposition A subspace A a secondcountablespace is also second countable

Def Mrt is a Hausdorff space if thqEM peg 7 disjoint open sets and V s t PE U and qe V

Proposition Any subspace of a Hausdorff spaceis Hausdorff

Hausdorff Proposition x one closed in spaces

MEAD open Backto Manifolds Topol ogical Manifolds

et Mit be a topological space Suppose Mis Hausdorff andsecond countable

Te c hn i c a lre a s o n g i 1 Partition ofunity 2 Compact exhaustion 3 Existenceof a Riemannianmeth 4 Embedding theorems

Dedi Mit is locally Euclidean dim m if C IRM open t PEM 3 U neighbd of P and F V is a homeomorphism and 70 U V that vern

fcq Cx i'm

x q XmCq

Xi U IR coordinates of geo IMPORTANT TERMINOLOGY

Vid is called a coordinate chat

COFFIN U is called a coordinateneighbnd of P

is called a coordinate map coordinate system

equivalent to M is locally homeomorphic to Rm

DeffTopologicalManifat

M is a topological manifold of dim rn if it's a Hausdorff second countable topological space that is locally Euclidean f din m

Is the dimension well defined amount

M alum

Tait

i 012 Uz zooli 01 n ERM EIR drool CU AUD RIGAUD

homeomorphism theorem Invariance dimension If UEIRM and VERN are opensets that are homeomorphic then man Needs the toolsof AlgebraicTo p o l o g y

beyondthecourse

Definition of smooth Manifold

Let M be a topological manifold ofdim n

Consider two coordinate charts UN and CU 4

transitionmap n called DEIR homeomorphism g u 4am i 2 oleum n C IRn E IR Det i 2 charts Uol and CV 4 are C compatible if the transitionmap 40 is a diffeomorphism

4051 Dam Kuru anti econ acorns

Det A atlas is a collection of charts manifold A Va Ola covering the

and are pairwise co compatible

precisely 1 Y U M 2 Varda and UB Op on copmpatible tap

Question Is C compatible an equivalence relation Reflexive Symmetric transitive Suppose Ul Q is compatible with Uz oh Uz Az is compatible with 03,03

question is CU oh compatible with Us.dz more precisely i

is also idiots 013 Us

a diffeomorphism U U AUS

Akira's Answer whereisthisthe it'sonline on 01 01300i Hopi µ Giffen 4 differ

3001 is a diffeomorphism

013cm i dillies all we can say 3001 is a chiffon Nat transitive 110 EIU fledEU EV C u Example g S uses c

S ee c i

2K gotta i U Co 21T Gosa Sino O 0 Yi V C IT TI cos 0 Sino is ELTIM

0104 i 4 UN Kono diffeomorphism i show it's a

a Thismakes Chol CU 413 atlas far s Detfasmathmaniff

Asmooth manifold Adimn is a topological manifold of dimn equipped with a C Atlas

The atlas gives thetopological manifold a Smooth structure makingit a smooth manifold