If from the real line ℝ, the irrational numbers are dropped, it becomes a line with holes which is what we call the subspace of rational numbers. Intuitively, the absence of holes in the real line makes it a single piece while rationals are not a single piece. If we drop the origin (0, 0) from the 2-dimensional Euclidean space ℝ2, it still remains a single piece while the removal of the y-axis will make two pieces of ℝ2, viz. the right half plane and the left half plane. The property of being a single piece is very important in mathematical analysis, and is known as connectedness. All intervals in the real line ℝ share this property. Thus, the real line is connected while the set of rational numbers is not. The metric spaces ℝ2, ℝ2 - {(0, 0)} are connected while ℝ2 minus y-axis is not connected. It will be seen that the famous intermediate value theorem: If f : [a, b] → ℝ is a continuous function, and f(a) ≤ c ≤ f(b), then there exists a x in [a, b] such that f(x) = c, depends not only on the continuity of f but also on the property of connectedness of the interval [a, b]. Though the property of connectedness is quite intuitive, it requires some effort to formulate it precisely in mathematical terms, and so the reader is advised to have a little patience.

Let X be a set, and f : X → X be a mapping. A point x in X is said to be a fixed point of f if f(x) = x. There are a number of problems in mathematics whose solutions are fixed points of certain mappings. Sometimes it is not possible to find a fixed point of a mapping exactly. Approximation methods are applied to get a value which is near to the required fixed point. Newton's method in numerical analysis can be considered such an approximation method. Some of these problems will be discussed in this chapter.

Fixed Point Theorems

Let X be a set, and f : X → X be a mapping. Then f need not have a fixed point. However, if we impose a certain mathematical structure on X and f, then f may have a fixed point. Our first attempt in this direction is the following.

Theorem 9.1.1 Every continuous function f : [a, b] → [a, b] has atleast one fixed point.

Proof. If a or b is a fixed point of f, then there is nothing to prove. So we assume that f(a) ≠ a and f(b) ≠ b.

Let us define a function g : [a, b] → [a, b] by setting g(x) = f(x) - x. Then g is obviously a continuous function. Since f(a) is in [a, b], f(a) > a. Similarly, f(b) < b. This means that g(a) > 0, and g(b) < 0.

Most working mathematicians agree that the collections whose members are numbers (of some fixed sort) are sets: and further the sets whose members are such sets, and the sets of those etc. This kind of argument thinks sets as cumulative type structures in which we begin by recognising some simple type of collections as sets and then construct from these in stages more and more complex type of sets. The following collections are sets.

Level 0: Some individuals (about which we presuppose nothing).

Level 1: All collections whose members are individuals.

Level 2: All collections whose members are in level 0 or 1.

At each level we take all collections whose members are in earlier levels. Thus, if the individuals are natural numbers, the collection {1, 2} occurs at level 1, the collection {1, {2}} occurs are level 2. {1, {2}, {1, {2}}} at level 3, and so on.

Note that every set occurs at every level after the one at which it is introduced, and it is taken the same at every occurrence. Two obvious questions arise with this definition. What shall we take as individuals? And, how far do the levels go? A cumulative type theory of sets has been developed by taking no individuals at all at level 0. (This is, in fact equivalent to taking the empty set as the only set at level 1).

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Definition and Examples of Metric Spaces

In ancient times, when man learnt to count his sheeps, at the same time, for the fear of losing the sheeps in the wild in dark, he, probably, have thought how far the sheeps might have gone from his hutments. Ever since the concept of distance and its measurement has puzzled the man. However, the effort put in to understand these, have paid rich dividends. The most basic definitions in mathematics were initially understood and described using the concept of a distance. For instance, the concepts such as the limit of a function, the convergence of a sequence and continuity of a function at a point used the notion of distance for their description. The definition that follows captures the most fundamental features of the concept of distance as we know it in our daily life. These are:

1. The distance between two objects is greater than or equal to zero.

2. The distance between two objects is zero if and only if the two objects are same.

3. The distance of any object x from another object y is same as the distance of y from x.

4. If there are three objects x, y and z, then the distance between x and y is less than or equal to the sum of the distance between x and z and the distance between y and z.

Real analysis is the branch of mathematical analysis which is primarily concerned with limit processes and continuity of real valued functions defined on the subsets of the real line. The considerations of these notions have led mathematicians to generalise these in various ways, for example, to study the convergence of sequences of real or complex numbers, the continuity of complex valued functions defined over the domains of complex numbers, etc. The reader who has done a course of real analysis must have noted that these concepts of limit, continuity of a function and convergence of sequence depend for their meaning on the notion of the absolute value of the difference of two numbers when they are regarded as the points on the real line, which is nothing but the distance between the points. It is no exaggeration if one says that the notion of distance permeates everywhere in the entire course of real analysis and also in geometry.

When the above notion of distance (or metric) for real numbers is extended to abstract sets in general, then the resulting mathematical structures are known as metric spaces. The metric spaces were first introduced by Maurice Fréchet in his thesis for doctorate in mathematics in 1906 and are being investigated since then. The metric spaces provide an appropriate setting to study ‘closeness’ of objects by defining a distance between them. This notion of distance captures the semantics of the real life usage of the meaning of the word ‘distance’.

SECTION-1: Growth of Hindu Mahasabha and Its Emergence as an All-India Force, 1938–1941

After the 1937 elections, the Congress swept the polls with respect to the Hindu seats. The Muslim League did not cut much ice in Bengal, as elsewhere, and Fazlul Haq's Krishak Praja Party came to the Bengal Legislature with a fairly large representation, with Haq defeating Nazimuddin at Patuakhali; and “it looked as if the reactionary Muslim elements had received a death blow”. If the Congress had responded to Haq's entreaty to form a coalition ministry in Bengal with himself as Premier, Bengal “would have developed into a strong and healthy province with the common efforts of representative Hindus and Muslims”. But as Dr Mookerjee says, “Congress policy regarding acceptance of office was still uncertain. Sarat Bose himself was even then deluding himself with such a declaration, ‘The Congress has come to the Legislature to end the Constitution and not to work it.’… It (Congress) was not prepared to co-operate or form a coalition with anyone. It must rule as a single party.” So the Congress High Command did not allow the Bengal Congress to join with Fazlul Haq's party, thus throwing Bengal “at the mercy of British-cum-League domination”. Accordingly, Haq joined the League in forming a ministry on 1 April 1937, headed by himself. He inducted N.R. Sarkar, B.P. Singh Roy and Maharaja of Cossimbazar as the Caste Hindu ministers.

SECTION-1: Interregnum between Bengal Ministership and Central Ministership

Syama Prasad's resignation made the fall of the Haq Ministry inevitable. When the Budget for the year 1943 was introduced by Fazlul Haq, who had taken over the Finance portfolio following Syama Prasad's resignation, it was apparent that the ministry would face quite a few challenges during the budget session. The strength of the Opposition was increasing daily. The League and the Europeans had combined, and a number of Muslims and Scheduled Caste members joined the Opposition after deserting the ministerial party. The official Congress party continued to attack the Ministry and made it clear that its continuance depended on its charity. The Communist party was carrying on a persistent propaganda against the Ministry and it joined hands with the League for this purpose. Governor Herbert urged Haq to bring in the Muslim League. A coalition with the League was acceptable to the truncated Ministry as well as to Syama Prasad who had always been prepared “to have a ministry representing all parties”. But the League “Maintained that while it was prepared to cooperate with any Non-Muslim party, it would never join any ministry along with the non-Muslim League Muslims”. This was in accordance with Jinnah's contention that the Muslim League was the sole representative organisation of the Muslims of India and the non-League Muslims were untouchables. Just before the Budget proceedings were about to be concluded, Haq was summered by the Governor on the evening of 28 March.

Programme of Bharatiya Jana Sangh

JANA SANGH PEOPLE'S PARTY PROGRAMME (APPROVED BY A CONFERENCE OF ORIGINAL MEMBERS OF THE PARTY)

1. The Party will give its first and foremost attention to the supply of food and cloth at reasonable prices. It will also be its immediate endeavour to secure reductions in the prices of other essential commodities. Unemployment will also receive its earnest attention. In carrying on its activities, the Party will try to create an enthusiasm in the heart of the nation and adopt a programme which will inspire all sections of the people to work for a common national purpose.

2. While not ignoring the beneficial effects on the country of long-term schemes like river-vally projects and reclamation of land, immediate attention will have to be devoted to increasing the production of food-crops and other agricultural commodities from existing acreage of land. It will be possible to raise the yeield per acre through smallscale irrigation schemes and the more plentiful supply of seeds, manure and implements. Uncultivated land will be requisitioned wherever necessary and co-operative farming will be encouraged.

3. The party will advocate a more rational procurement polity which will instead of impeding help production.

4. The Party believes that our economic salvation lies in increasing production and in adopting a fair and more equitable system of distribution. The party's activities will be constantly and considerably directed towards this end.

5. The Party will work for planned decentralisation of industries and the development of small and medium-size industries aiming at regional self-sufficiency as far as possible and combating unemployment, particularly amongst the poor and middle class people.

6. […]