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4.8: Continuity on Compact Sets. Uniform Continuity


I. Some additional important theorems apply to functions that are continuous on a compact set (see §6).

Theorem (PageIndex{1})

If a function (f : A ightarrowleft(T, ho^{prime} ight), A subseteq(S, ho),) is relatively continuous on a compact set (B subseteq A,) then (f[B]) is a compact set in (left(T, ho^{prime} ight) .) Briefly,

[ ext{the continuous image of a compact set is compact.}]

Proof

To show that (f[B]) is compact, we take any sequence (left{y_{m} ight} subseteq f[B]) and prove that it clusters at some (q in f[B]).

As (y_{m} in f[B], y_{m}=fleft(x_{m} ight)) for some (x_{m}) in (B .) We pick such an (x_{m} in B) for each (y_{m},) thus obtaining a sequence (left{x_{m} ight} subseteq B) with

[fleft(x_{m} ight)=y_{m}, quad m=1,2, ldots]

Now by the assumed compactness of (B,) the sequence (left{x_{m} ight}) must cluster at some (p in B .) Thus it has a subsequence (x_{m_{k}} ightarrow p .) As (p in B,) the function (f) is relatively continuous at (p) over (B) (by assumption). Hence by the sequential criterion ((§ 2), x_{m_{k}} ightarrow p) implies (fleft(x_{m_{k}} ight) ightarrow f(p) ;) i.e.,

[y_{m_{k}} ightarrow f(p) in f[B].]

Thus (q=f(p)) is the desired cluster point of (left{y_{m} ight} . square)

This theorem can be used to prove the compactness of various sets.

Example (PageIndex{1})

(1) A closed line segment (L[overline{a}, overline{b}]) in (E^{n}left(^{*} ext { and in other normed spaces } ight)) is compact, for, by definition,

[L[overline{a}, overline{b}]={overline{a}+t vec{u} | 0 leq t leq 1}, ext{ where } vec{u}=overline{b}-overline{a}.]

Thus (L[overline{a}, overline{b}]) is the image of the compact interval ([0,1] subseteq E^{1}) under the (operatorname{map} f : E^{1} ightarrow E^{n},) given by (f(t)=overline{a}+t vec{u},) which is continuous by Theorem 3 of §3. (Why?)

(2) The closed solid ellipsoid in (E^{3},)

[left{(x, y, z) | frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}+frac{z^{2}}{c^{2}} leq 1 ight},]

is compact, being the image of a compact globe under a suitable continuous map. The details are left to the reader as an exercise.

lemma (PageIndex{1})

Every nonvoid compact set (F subseteq E^{1}) has a maximum and a minimum.

Proof

By Theorems 2 and 3 of §6, (F) is closed and bounded. Thus (F) has an infimum and a supremum in (E^{1}) (by the completeness axiom), say, (p=inf F) and (q=sup F .) It remains to show that (p, q in F .)

Assume the opposite, say, (q otin F .) Then by properties of suprema, each globe (G_{q}(delta)=(q-delta, q+delta)) contains some (x in B) (specifically, (q-delta

[(forall delta>0) quad F cap G_{ eg q}(delta) eq emptyset;]

i.e., (F) clusters at (q) and hence must contain (q) (being closed). However, since (q otin F,) this is the desired contradiction, and the lemma is proved. (square)

The next theorem has many important applications in analysis.

Theorem (PageIndex{2})

(Weierstrass).

(i) If a function (f : A ightarrowleft(T, ho^{prime} ight)) is relatively continuous on a compact set (B subseteq A,) then (f) is bounded on (B ;) i.e., (f[B]) is bounded.

(ii) If, in addition, (B eq emptyset) and (f) is real (left(f : A ightarrow E^{1} ight),) then (f[B]) has a maximum and a minimum; i.e., f attains a largest and a least value at some points of (B).

Proof

Indeed, by Theorem (1, f[B]) is compact, so it is bounded, as claimed in ((i)).

If further (B eq emptyset) and (f) is real, then (f[B]) is a nonvoid compact set in (E^{1},) so by Lemma (1,) it has a maximum and a minimum in (E^{1} .) Thus all is proved. (square)

Note 1. This and the other theorems of this section hold, in particular, if (B) is a closed interval in (E^{n}) or a closed globe in (E^{n}left(^{*} ext { or } C^{n} ight)) (because these sets are compact - see the examples in §6). This may fail, however, if (B) is not compact, e.g., if (B=(overline{a}, overline{b}) .) For a counterexample, see Problem 11 in Chapter 3, §13.

Theorem (PageIndex{3})

If a function (f : A ightarrowleft(T, ho^{prime} ight), A subseteq(S, ho)), is relatively continuous on a compact set (B subseteq A) and is one to one on (B) (i.e., when restricted to (B)), then its inverse, (f^{-1}), is continuout on (f[B]).

Proof

To show that (f^{-1}) is continuous at each point (q in f[B]), we apply the sequential criterion (Theorem 1 in §2). Thus we fix a sequence (left{y_{m} ight} subseteq f[B], y_{m} ightarrow q in f[B]), and prove that (f^{-1}left(y_{m} ight) ightarrow f^{-1}(q)).

Let (f^{-1}left(y_{m} ight)=x_{m}) and (f^{-1}(q)=p) so that

[y_{m}=fleft(x_{m} ight), q=f(p), ext { and } x_{m}, p in B.]

We have to show that (x_{m} ightarrow p), i.e., that

[(forall varepsilon>0)(exists k)(forall m>k) quad holeft(x_{m}, p ight)

Seeking a contradiction, suppose this fails, i.e., its negation holds. Then (see Chapter 1, §§1–3) there is an (epsilon > 0) such that

[(forall k)left(exists m_{k}>k ight) quad holeft(x_{m_{k}}, p ight) geq varepsilon,]

where we write “(m_{k})” for “(m)” to stress that the (m_{k}) may be different for different (k). Thus by (1), we fix some (m_{k}) for each (k) so that (1) holds, choosing step by step,

[m_{k+1}>m_{k}, quad k=1,2, ldots]

Then the (x_{m_{k}}) form a subsequence of ({x_{m}}), and the corresponding (y_{m_{k}}=f(x_{m_{k}})) form a subsequence of (left{y_{m} ight}). Henceforth, for brevity, let (left{x_{m} ight}) and (left{y_{m} ight}) themselves denote these two subsequences. Then as before, (x_{m} in B, y_{m}=fleft(x_{m} ight) in f[B]), and (y_{m} ightarrow q, q=f(p)). Also,by(1),

[(forall m) quad holeft(x_{m}, p ight) geq varepsilonleft(x_{m} ext { stands for } x_{m_{k}} ight).]

Now as (left{x_{m} ight} subseteq B) and (B) is compact, (left{x_{m} ight}) has a (sub)subsequence

[x_{m_{i}} ightarrow p^{prime} ext { for some } p^{prime} in B.]

As (f) is relatively continuous on (B), this implies

[fleft(x_{m_{i}} ight)=y_{m_{i}} ightarrow fleft(p^{prime} ight)]

However, the subsequence (left{y_{m_{i}} ight}) must have the same limit as (left{y_{m} ight}), i.e., (f(p)). Thus (fleft(p^{prime} ight)=f(p)) whence (p=p^{prime}) (for (f) is one to one on (B)), so (x_{m_{i}} ightarrow p^{prime}=p).

This contradicts (2), however, and thus the proof is complete. (square)

Example (PageIndex{2})

(3) For a fixed (n in N,) define (f :[0,+infty) ightarrow E^{1}) by

[f(x)=x^{n}.]

Then (f) is one to one (strictly increasing) and continuous (being a monomial; see §3). Thus by Theorem 3, (f^{−1}) (the nth root function) is relatively continuous on each interval

[f=[a^{n}, b^{n}].]

hence on ([0,+infty).)

See also Example (a) in §6 and Problem 1 below.

II. Uniform Continuity. If (f) is relatively continuous on (B), then by definition,

[(forall varepsilon>0)(forall p in B)(exists delta>0)left(forall x in B cap G_{p}(delta) ight) quad ho^{prime}(f(x), f(p))

Here, in general, (delta) depends on both (epsilon) and (p) (see Problem 4 in §1); that is, given (epsilon > 0), some values of (delta) may fit a given p but fail (3) for other points.

It may occur, however, that one and the same (delta) (depending on (epsilon) only) satisfies (3) for all (p in B) simultaneously, so that we have the stronger formula

[(forall varepsilon>0)(exists delta>0)(forall p, x in B | ho(x, p)

Definition

If (4) is true, we say that (f) is uniformly continuous on (B).

Clearly, this implies (3), but the converse fails.

Theorem (PageIndex{4})

If a function (f : A ightarrowleft(T, ho^{prime} ight), A subseteq(S, ho)), is relatively continuous on a compact set (B subset A), then (f) is also uniformly continuous on (B).

Proof

(by contradiction). Suppose (f) is relatively continuous on (B), but (4) fails. Then there is an (epsilon > 0) such that

[(forall delta>0)(exists p, x in B) quad ho(x, p)

here (p) and (x) on (delta). We fix such an (epsilon) and let

[delta=1, frac{1}{2}, ldots, frac{1}{m}, dots]

Then for each (delta) (i.e., each (m)), we get two points (x_{m}, p_{m} in B) with

[ holeft(x_{m}, p_{m} ight)

and

[ ho^{prime}left(fleft(x_{m} ight), fleft(p_{m} ight) ight) geq varepsilon, quad m=1,2, ldots]

Thus we obtain two sequences, (left{x_{m} ight}) and (left{p_{m} ight}), in (B). As (B) is compact, (left{x_{m} ight}) has a subsequence (x_{m_{k}} ightarrow q(q in B)). For simplicity, let it be (left{x_{m} ight}) itself; thus

[x_{m} ightarrow q, quad q in B.]

Hence by (5), it easily follows that also (p_{m} ightarrow q) (because ( holeft(x_{m}, p_{m} ight) ightarrow 0). By the assumed relative continuity of (f) on (B), it follows that

[fleft(x_{m} ight) ightarrow f(q) ext { and } fleft(p_{m} ight) ightarrow f(q) ext { in }left(T, ho^{prime} ight).]

This, in turn, implies that ( ho^{prime}left(fleft(x_{m} ight), fleft(p_{m} ight) ight) ightarrow 0), which is impossible, in view of (6). This contradiction completes the proof. (square)

Example (PageIndex{1})

(a) A function (f : A ightarrow left( T , ho ^ { prime } ight) , A subseteq ( S , ho )), ic called a contraction map (on (A)) iff

[ ho ( x , y ) geq ho ^ { prime } ( f ( x ) , f ( y ) ) ext { for all } x , y in A.]

Any such map is uniformly continuous on A. In fact, given (varepsilon > 0), we simply take (delta = varepsilon). Then ( forall x , p in A )

[ ho ( x , p ) < delta ext { implies } ho ^ { prime } ( f ( x ) , f ( p ) ) leq ho ( x , p ) < delta = varepsilon,]

as required in (3).

(b) As a special case, consider the absolute value map (norm map) given by

[f ( overline { x } ) = | overline { x } | ext { on } E ^ { n } left( ^ { * } ext { or another normed space } ight).]

It is uniformly continuous on(E^{n}) because

[| | overline { x } | - | overline { p } | | leq | overline { x } - overline { p } | , ext { i.e., } ho ^ { prime } ( f ( overline { x } ) , f ( overline { p } ) ) leq ho ( overline { x } , overline { p } ),]

which shows that (f) is a contraction map, so Example (a) applies.

(c) Other examples of contraction maps are

(1) constant maps (see §1, Example (a)) and

(2) projection maps (see the proof of Theorem 3 in §3).

Verify!

(d) Define (f : E ^ { 1 } ightarrow E ^ { 1 }) by

[f ( x ) = sin x]

By elementary trigonometry, (| sin x | leq | x |). Thus (left( forall x , p in E ^ { 1 } ight))

[egin{aligned} | f ( x ) - f ( p ) | & = | sin x - sin p | & = 2 left| sin frac { 1 } { 2 } ( x - p ) cdot cos frac { 1 } { 2 } ( x + p ) ight| & leq 2 left| sin frac { 1 } { 2 } ( x - p ) ight| & leq 2 cdot frac { 1 } { 2 } | x - p | = | x - p | end{aligned},]

and (f) is a contraction map again. Hence the sine function is uniformly continuous on (E^{1}); similarly for the cosine function.

(e) Given ( emptyset eq A subseteq ( S , ho ) , ) define ( f : S ightarrow E ^ { 1 } ) by

[
f ( x ) = ho ( x , A ) ext { where } ho ( x , A ) = inf _ { y in A } ho ( x , y )
]

It is easy to show that

[
( forall x , p in S ) quad ho ( x , A ) leq ho ( x , p ) + ho ( p , A )
]

i.e.,

[
f ( x ) leq ho ( p , x ) + f ( p ) , ext { or } f ( x ) - f ( p ) leq ho ( p , x )
]

Similarly, ( f ( p ) - f ( x ) leq ho ( p , x ) . ) Thus

[
| f ( x ) - f ( p ) | leq ho ( p , x )
]

i.e., ( f ) is uniformly continuous (being a contraction map).

(f) The identity map ( f : ( S , ho ) ightarrow ( S , ho ) , ) given by

[
f ( x ) = x
]

is uniformly continuous on ( S ) since

[
ho ( f ( x ) , f ( p ) ) = ho ( x , p ) ext { (a contraction map!) }
]

However, even relative continuity could fail if the metric in the domain space ( S ) were not the same as in ( S ) when regarded as the range space (e.g., make ( ho ^ { prime } ) discrete!)

(g) Define ( f : E ^ { 1 } ightarrow E ^ { 1 } ) by

[
f ( x ) = a + b x quad ( b eq 0 ).
]

Then

[
left( forall x , p in E ^ { 1 } ight) quad | f ( x ) - f ( p ) | = | b | | x - p |;
]

i.e.,

[
ho ( f ( x ) , f ( p ) ) = | b | ho ( x , p ).
]

Thus, given ( varepsilon > 0 , ) take ( delta = varepsilon / | b | . ) Then

[
ho ( x , p ) < delta Longrightarrow ho ( f ( x ) , f ( p ) ) = | b | ho ( x , p ) < | b | delta = varepsilon,
]

proving uniform continuity.

(h) Let

[
f ( x ) = frac { 1 } { x } quad ext { on } B = ( 0 , + infty ).
]

Then ( f ) is continuous on ( B , ) but not uniformly so. Indeed, we can prove the negation of ( ( 4 ) , ) i.e.

[
( exists varepsilon > 0 ) ( forall delta > 0 ) ( exists x , p in B ) quad ho ( x , p ) < delta ext { and } ho ^ { prime } ( f ( x ) , f ( p ) ) geq varepsilon.
]

Take ( varepsilon = 1 ) and any( delta > 0 . ) We look for ( x , p ) such that

[
| x - p | < delta ext { and } | f ( x ) - f ( p ) | geq varepsilon,
]

i.e.,

[
left| frac { 1 } { x } - frac { 1 } { p } ight| geq 1,
]

This is achieved by taking

[
p = min left( delta , frac { 1 } { 2 } ight) , x = frac { p } { 2 } . quad ( ext { Verify! } )
]

Thus ( ( 4 ) ) fails on ( B = ( 0 , + infty ) , ) yet it holds on ( [ a , + infty ) ) for any ( a > 0 ) .
(Verify!)


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The content of the first-year courses in the Bachelor program in Mathematics. In particular, each student is expected to be familiar with notion of continuity for functions from/to Euclidean spaces, and with the content of the corresponding basic theorems (Bolzano, Weierstrass etc..). In addition, some degree of scientific maturity in writing rigorous proofs (and following them when presented in class) is absolutely essential.

Content

An introduction to topology i.e. the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures.

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Terminology and some auxiliary results

Notation

All topological spaces under consideration are assumed to be Tikhonov.

We say that a family (mathcal ) of sets has order (le n) if every subfamily (mathcal subset mathcal ) of cardinality (n+2) has an empty intersection (in other terminology, the family (mathcal ) is point- ((n+1)) ). The family (mathcal ) has finite order if it has order (le n) for some (nin omega ) .

The family (mathcal ) of subsets of a space X is (T_0) -separating if, for every pair of distinct points x, y of X, there is (Uin mathcal ) containing exactly one of the points x, y.

For a locally compact space X, (alpha (X)) denotes the one point compactification of X. We denote the point at infinity of this compactification by (infty _X) .

Function spaces

Given a compact space K, by C(K) we denote the Banach space of continuous real-valued functions on K, equipped with the standard supremum norm.

Scattered spaces

A space X is scattered if no nonempty subset (Asubseteq X) is dense-in-itself.

For a scattered space K, by Cantor–Bendixson height ht(X) of K we mean the minimal ordinal (alpha ) such that the Cantor–Bendixson derivative (K^<(alpha )>) of the space K is empty. The Cantor–Bendixson height of a compact scattered space is always a nonlimit ordinal.

A surjective map (f:X ightarrow Y) between topological spaces is said to be irreducible if no proper closed subset of X maps onto Y. If X is compact, by Kuratowski–Zorn Lemma, for any surjective map (f:X ightarrow Y) , there is a closed subset (Csubseteq X) such that the restriction (fupharpoonright C) is irreducible.

The following facts concerning continuous maps of scattered compact spaces are well known, cf. the proof of Proposition 8.5.3 and Exercise 8.5.10(C) in [16].

Proposition 2.1

Let K be a scattered compact space and let (varphi : K ightarrow L) be a continuous surjection. Then, for each ordinal (alpha ) , we have (L^ <(alpha )>subseteq varphi (K^<(alpha )>)) . In particular, (ht(L) le ht(K)) .

Proposition 2.2

Let K be a scattered compact space and let (varphi : K ightarrow L) be a continuous irreducible surjection. Then (L' = varphi (K')) and (varphi upharpoonright (K K')) is a bijection onto (L L') .

Eberlein and Corson compact spaces

A space K is an Eberlein compact space if K is homeomorphic to a weakly compact subset of a Banach space. Equivalently, a compact space K is an Eberlein compactum if K can be embedded in the following subspace of the product (mathbb ^Gamma ) :

If K is homeomorphic to a weakly compact subset of a Hilbert space, then we say that K is a uniform Eberlein compact space. All metrizable compacta are uniform Eberlein.

A compact space K is Corson compact if, for some set (Gamma ) , K is homeomorphic to a subset of the (Sigma ) -product of real lines

Clearly, the class of Corson compact spaces contains all Eberlein compacta.

Spaces (sigma _(X))

Given a set (Gamma ) and (nin omega ) , by (sigma _(Gamma )) we denote the subspace of the product (2^Gamma ) consisting of all characteristic functions of sets of cardinality (le n) . The space (sigma _(Gamma )) is uniform Eberlein and scattered of height (n+1) .

For (Ain [Gamma ]^) , we denote the standard clopen neighborhood ((Gamma ): Asubset B>) of (chi _A) in (sigma _(Gamma )) by (V_) .

To simplify the notation we will say that a compact space K belongs to the class (mathcal _n) if K can be embedded in the space (sigma _(Gamma )) for some set (Gamma ) . We will denote the union (igcup _mathcal _n) by (mathcal _<<omega >) . Trivially, if a compact space K belongs to any of the above classes, then each closed subset of K is also in the same class. One can also easily verify that the class (mathcal _<<omega >) is preserved under taking finite products, cf. [1, p. 148].

Proposition 2.3

For a compact space K and (nin omega ) , the following conditions are equivalent:

K has a (T_0) -separating point-n family of clopen subsets

K belongs to the class (mathcal _n) .

Proof

((ii) (Rightarrow ) (i)) Suppose that K is a subspace of the space (sigma _(Gamma )) for some set (Gamma ) . For (gamma in Gamma ) , let (U_gamma = ) . One can easily verify that the family () is a (T_0) -separating point-n family of clopen subsets of K. (square )

Lemma 2.4

Let K be an infinite compact subset of (sigma _(Gamma )) for some set (Gamma ) and (nin omega ) . Then K can can be embedded into (sigma _(kappa )) , where (kappa ) is the weight w(K) of K.

Proof

Follows from the proof of Lemma 2.3 and the well known fact that, for an infinite compact space the cardinality of the family of clopen subsets of K is bounded by w(K). (square )

Lemma 2.5

Let (Gamma ) be an infinite set. Then for any (n,kin omega , kge 1) , the discrete union of k copies of (sigma _(Gamma )) embeds into (sigma _(Gamma )) .

Proof

Let (X=>) be a set disjoint with (Gamma ) . For (fin sigma _(Gamma )) and (i< k) let (f_i: Gamma cup X ightarrow 2) be defined by

One can easily verify that, if we assign to a function f from i-th copy of (sigma _(Gamma )) , the function (f_i) , then we will obtain an embedding of the discrete union of k copies of (sigma _(Gamma )) into (sigma _(Gamma cup X)) , a copy of (sigma _(Gamma )) . (square )

Theorem 2.6

(Argyros and Godefroy) Every Eberlein compactum K of weight (<omega _omega ) and of finite height belongs to the class (mathcal _<<omega >) .

Example 2.7

(Bell and Marciszewski [4]) There exists an Eberlein compactum K of weight (omega _omega ) and height 3 which does not belong to (mathcal _<<omega >) .

Luzin sets and its variants

Usually, a subset L of real line (mathbb ) is called a Luzin set if X is uncountable and, for every meager subset A of (mathbb ) the intersection (Acap L) is countable. Let (kappa le lambda ) be uncountable cardinal numbers. We will say that a subset L of a Polish space X is a ((lambda ,kappa )) -Luzin set if X has the cardinality (lambda ) , and, for every meager subset A of X the intersection (Acap L) has the cardinality less than (kappa ) . In this terminology, the existence of a Luzin set in (mathbb ) is equivalent to the existence of a ((omega _1,omega _1)) -Luzin set.

Since, for every Polish space X without isolated points there is a Borel isomorphism (h:X ightarrow mathbb ) such that (Asubseteq X) is meager if and only if, h(A) is meager in (mathbb ) , it follows that the existence of a ((lambda ,kappa )) -Luzin set in such X is equivalent to the existence of a ((lambda ,kappa )) -Luzin set in (mathbb ) .

It is known that, for each (nge 1) the existence of a ( (omega _n,omega _1)) -Luzin set in (mathbb ) is consistent with ZFC , cf. [2, Lemma 8.2.6].

Cardinal numbers (mathfrak ) and (mathrm (mathcal ))

Recall that the preorder (le ^*) on (omega ^omega ) is defined by (fle ^* g) if (f(n)le g(n)) for all but finitely (nin omega ) . A subset A of (omega ^omega ) is called unbounded if it is unbounded with respect to this preorder. In Sect. 4, we will use two cardinal numbers related with the structure of the real line

It is well known that (mathfrak le mathrm (mathcal )) (cf. [2, Ch. 2]), and, for each natural number (nge 1) , the statement (mathfrak = omega _n) is consistent with ZFC , (cf. [17, Theorem 5.1]).

Aleksandrov duplicate AD(K) of a compact space K

Recall the construction of the Aleksandrov duplicate AD(K) of a compact space K.

(AD(K) = K imes 2) , points (x, 1), for (xin K) , are isolated in AD(K) and basic neighborhoods of a point (x, 0) have the form ((U imes 2) <(x,1)>) , where U is an open neighborhood of x in K.

The following fact is well known (cf. [10]).

Proposition 2.8

The Aleksandrov duplicate AD(K) of an (uniform) Eberlein compact space K is (uniform) Eberlein compact.

Proof

Without loss of generality we can assume that K is a subspace of (c_0(Gamma )) ( (ell _2(Gamma )) ), equipped with the pointwise topology, for some set (Gamma ) . We will show that AD(K) can be embedded into the space (c_0(Gamma cup K)) ( (ell _2(Gamma cup K)) ). For (xin K) and (i=0,1) define a function (f_:Gamma cup K ightarrow mathbb ) as follows:

One can easily verify that the mapping ((x,i)mapsto f_) gives the desired embedding. (square )


Panjab University M.Sc. Mathematics 2020-21 Syllabi : puchd.ac.in

(ii) Sequences and series: Convergent sequences (in metric spaces). Subsequences. Cauchy sequences. Upper and lower limits of a sequence of real numbers. Riemann’s Theorem on Rearrangements of series of real and complex numbers.

(iii) Continuity: Limits of functions (in metric spaces). Continuous functions. Continuity and compactness. Continuity and connectedness. Monotonic functions.

Unit- II:
(iv) The Riemann-Stieltjes integral: Definition and existence of the Riemann-Stieltjes integral. Properties of the integral. Integration of vector-valued functions. Rectifiable curves.

(v) Sequences and series of functions: Problem of interchange of limit processes for sequences of functions. Uniform convergence. Uniform convergence and continuity. Uniform convergence and integration. Uniform convergence and differentiation. Equicontinuous families of functions, The Stone- Weierstrass theorem.

Math 602S: Algebra- I
Unit- I:
Review of basic concepts of groups with emphasis on exercises. Permutation groups, Even and odd permutations, Conjugacy classes of permutations, Alternating groups, Simplicity of An, n > 4. Cayley’s Theorem, Direct products, Fundamental Theorem for finite abelian groups, Sylow theorems and their applications, Finite Simple groups [Scope as in chapters 2-4 Modern Algebra by Surjeet Singh and Qazi Zameerudin, Eighth Edition and chapters 11, 24, 25 of Contemporary Abstract Algebra by Gallian, Fourth Edition]

Unit-II:
Survey of some finite groups, Groups of order p2, pq (p and q primes). Solvable groups, Normal and subnormal series, composition series, the theorems of Schreier and Jordan Holder [Scope as in Chapters 6 of Modern Algebra by Surjeet Singh and Qazi Zameerudin, Eighth Edition and Chapter 7 of Algebra, Vol. I by Luther and Passi].

Review of basic concepts of rings with emphasis on exercises. Polynomial rings, formal power series rings, matrix rings, the ring of Guassian Integers. [Scope as in Chapters 7, 8 and 9 of Modern Algebra by Surjeet Singh and Qazi Zameerudin, Eighth Edition , 2006].

Math 603S: Differential Equations

Unit-I:
Differential Equations Existence and uniqueness of solution of first order equations. Boundary value problems and Strum-Liouville theory. ODE in more than 2-variables. [Scope as in Chapter V of the book ‘An introduction to Ordinary Differential Equations’ by E.A.Coddington and Chapters X & XI of the book ‘Elementary Differential Equations and Boundary Value Problems’ by W.E.Boyce and R.C.Diprima.]

Unit-II:
Partial differential equations of first order. Partial differential equations of higher order with constant coefficients. Partial differential equations of second order and their classification. [Scope as in Chapters I, II & III of the book ‘Elements of Partial Differential Equations’ by I.N.Sneddon].

Math 604S: Complex Analysis-I
Unit-I:
Complex plane, geometric representation of complex numbers, joint equation of circle and straight line, stereographic projection and the spherical representation of the extended complex plane. Topology on the complex plane, connected and simply connected sets. Complex valued functions and their continuity. Curves, connectivity through polygonal lines.Analytic functions, Cauchy-Riemann equations, Harmonic functions and Harmonic conjugates.Power series, exponential and trigonometric functions, arg z, log z, az and their continuous branches. (Scope as in “Foundations of Complex Analysis” by Ponnusamy S., Chapter 1, (§1.1-§ 1.5),Chapter 2 (§ 2.2, §2.3), Chapter 3, (§3.1-§3.5), Chapter 4, (§4.9).)

Unit-II:
Complex Integration, line integral, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disc, index of a point with respect to a closed curve, Cauchy’s integral formula, Higher derivatives, Morrera’s theorem, Liouville’s theorem, fundamental theorem of Algebra. The general form of Cauchy’s theorem. (Scope as in “Foundations of Complex Analysis” by Ponnusamy S., Chapter 4, (§4.1-§ 4.8), Chapter 6 (§ 6.4, §6.6).”Complex Analysis” by L/ V. Ahlfors, Chapter 4 (§1, 2, 4.1 to 4.5and §5.1)


Duality

I have already said that, given a set X with a quasi-uniformity U, seen with the induced topology, every compact saturated subset of X is closed in X –1 . This means that the cocompact topology on X is coarser than the topology of X –1 . When U is U0, the minimal compatible quasi-uniformity (see Proposition D), those two topologies coincide, as we now argue.

Proposition F. Let X be a locally compact topological space, and U be the minimal compatible quasi-uniformity U0. The topology induced by the dual quasi-uniformity U –1 on X coincides with the cocompact topology.

We expand the definition of R: R –1 [x] is the set of points y such that for every i, if yQi then xUi equivalently, such that for every i, if xUi then xQi in other words, it is the complement of QI, where QI is the union of the sets Qi, iI, and I is the collection of indices i such that xUi. QI is compact saturated, so its complement R –1 [x] is open in the cocompact topology. This complement contains x (R –1 [x] always contains x), and is included in O. This shows that O is an open neighborhood, in the cocompact topology, of each of its points, so that O is open in the cocompact topology.

Conversely, let O be any open subset of X in the cocompact topology. Its complement Q is compact saturated in X. By Lemma C, O is open in the topology induced by U –1 . ☐

We finally reach the result promised at the beginning of this post.

Theorem. Let X be a stably compact topological space. There is a unique quasi-uniformity U that induces the topology of X and such that the dual quasi-uniformity U –1 induces the cocompact topology, and this is the minimal compatible quasi-uniformity U0.

Proof. Existence is by Proposition F. In order to show uniqueness, we fix a quasi-uniformity U that induces the topology of X and such that the dual quasi-uniformity U –1 induces the cocompact topology. By Proposition D’, U contains U0, so we concentrate on showing the reverse inclusion.

Let R be any entourage of U. There is an entourage S in U such that S o SR. For every xX, it follows that S –1 [x] × S[x] is included in R: every pair (y,z) in S –1 [x] × S[x] is such that (y,x) ∈ S and (x,z) ∈ S, so (y,z) ∈ R. S –1 [x] is an open neighborhood of x in X d since U –1 induces the cocompact topology on X, and S[x] is an open neighborhood of x in X since U induces the original topology on X. Therefore R is an open neighborhood of (x,x) in X d × X, for every xX. In other words, R is an open neighborhood of (=) in X d × X. By Proposition E’, R in U0, and this finishes the proof. ☐


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