Neighbourhoods and Structure
Raise your arm, bring the tip of right hand's forefinger in your front and let it steadily approach the tip of your nose. Suppose it touches the nose and your finger sticks on; and you are not able to extricate your finger. You will take a lesson and never touch your nose again. Touching of the nose was indeed a crime: You are changing the surroundings, the neighbourhood of the finger's tip as well as that of your nose's tip. Punishment for the crime: Finger stuck with nose's tip. Enjoy your NEW STRUCTURE!
Observe carefully that neighbourhood of your finger's tip now include parts of your nose also; and nose's tip (becoming identical with finger's tip) has for its neighbourhood parts of your finger.
If the above rule and its punishments for violations are followed, clapping above your head in a PT class or clapping in applause can give you new structures (with glued hands).
Have you experienced a sting from a bee on your hand or on your cheek? One sided swollen face: a good case for the test of changed structure? The original point on the cheek gets an upward shift and corresponding shift get the neighbouring points. This change therefore does not change the neighbourhood structure: not a new structure according to our rule.
Have you seen a joker on a deflated balloon? Observe the joker on the inflated balloon. Repeat the experiment by making a small joker on your friend's cheek; and let bees visit this joker till you get a visibly enlarged joker. Observe that neighbourhood structure for no point has changed! So why should anybody mind!!
If one hand could be inflated or one finger pulled to $20$ cm; as long as neighbourhood of any of the point is not disturbed, a change of structure case will not be registered: Bending of the hand or finger, with pain or without it, will not make a case. (Not allowing bending to touch the shoulder or any other part of the body.)
Homeomorphisms
Observe the two structures: Change in neighbourhoods of fingertips or nose tip.
Observe carefully that neighbourhood of your finger's tip now include parts of your nose also; and nose's tip (becoming identical with finger's tip) has for its neighbourhood parts of your finger.
Changed Neighbourhoods for the terminal of the right hand (or left hand).
Have you experienced a sting from a bee on your hand or on your cheek? One sided swollen face: a good case for the test of changed structure? The original point on the cheek gets an upward shift and corresponding shift get the neighbouring points. This change therefore does not change the neighbourhood structure: not a new structure according to our rule.
Have you seen a joker on a deflated balloon? Observe the joker on the inflated balloon. Repeat the experiment by making a small joker on your friend's cheek; and let bees visit this joker till you get a visibly enlarged joker. Observe that neighbourhood structure for no point has changed! So why should anybody mind!!
If one hand could be inflated or one finger pulled to $20$ cm; as long as neighbourhood of any of the point is not disturbed, a change of structure case will not be registered: Bending of the hand or finger, with pain or without it, will not make a case. (Not allowing bending to touch the shoulder or any other part of the body.)
Distant and distinct points if got together and get glued, there is change of structure; and obviously so, its reverse cutting or tearing apart also changes structure.
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There is a popular paradox -- a puzzle for school children. Take two line segments of lengths of one unit and two units. Place them in a plane, in non-overlapping non-intersecting position. Join the end points of the two segments, by straights lines; and extend the lines till they intersect at point $O$.
Stretching a segment.
Join an arbitrary point $P$ of the first line segment to $O$ by a straight line. The straight line $OP$ meets the second line in one and only one point, say $P'$. On the other hand, if we choose an arbitrary point $Q'$ on the second line and join to $O$; then the line $OQ'$ meets the first line in a unique point $Q$. Thus, to each point in the first segment, there is one and only one point in the second segment. Hence the two line segments have the same length, implying $1 = 2$ !!
What is actually happening?
The correspondence between the segments is given by $F(t) = 2t$, as shown. The segment $[0,1]$ pulled becomes $[0,2]$ and observe that the neighbourhoods of the mapped point $P$ are exactly the image of the neighbourhoods of $P$ and the structure of $[0,1]$ is not changed.
The transformation (pulling or inflating) not disturbing the neighbourhoods (called homeomorphism) may change the size (measure). Structurally you have remained the same in ten years, but your size and volume is not the same. Is that a surprise!!
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New Structures
Let us take a piece of thread, with markings 0, 1, 2, 3, 4, 5 and 6 to fascinate discussion. We may lay it down on a table in several ways.
Threads with loose ends.
Observe the neighbourhoods of any two chosen points $x$ and $y$ on the thread. The neighbouring points of $x$ or that of $y$ remains unchanged, in the different settings of the thread. All of these therefore provide the same neighbourhood structure. Except the first, in all other settings the thread needs a plane (a two-dimensional space) to imbed. Yet, the structure in the first and the other cases is the same. It is easily seen that, a turn in the thread off the plane, can make it a three dimensional object; while its structure remains yet unaltered. (Nobody can stop our mind to provide it a turn to move in the fourth or higher dimension).
In all the above settings, we have restrained ourselves from bringing the points $6$ too close to $0$, fearing sticking and get glued. And if $6$ touches $0$, the forbidden happening; it results into the birth of a new structure, different from the earlier ones.
New Structures: Topologically or homeomorphically
identical.
identical.
The new structure requires the $0$ and $6$ to lose their independent identities, and the new point (we may call a coset) having for its neighbourhoods, both from the neighbourhoods of $0$ and neighbourhoods of $6$. The newborn is homeomorphically different from the earlier ones, also called topologically different. The three new-borns shown in the figure, though different from the earlier ones, are topologically or homeomorphically identical to each other; their neighbourhood structures being the same.
Again, we may begin with a rectangular sheet $-$ a subset of the Euclidean $\mathbb{R}^2$. Turn the sheet along the line $x = 1$.
A sheet and its folded form.
Observe the neighbourhoods of points on the original sheet and the neighbourhoods of the same points in the turned sheet (the turned-one being imbedded in $\mathbb{R}^3$).
Turning does not change the neighbourhoods of any of the points, and so the two structures are topologically identical. What happens if we turn the sheet along $x = 2$ also? Will it make a difference topologically if the turnings were circular?
Let us venture once again on the forbidden path. By a turning, we can glue together $C (3,2)$ and $D (0,2)$; also $A (0,0)$ and $B (3,0)$; and also $(0,4)$ to $(3,4)$ for each $0 < y < z$. Neighbourhoods structure has changed for so many points. And the new borns are given in the following Figure.
The two new structures are topologically identical.
The joining of points produced topologically new models called quotient spaces. It will be amusing to take five pieces (line-segments) $AB, CD, EF$, $GH$, and $KL$. Joining the points $A, B, C, D, E, F, G, H, K$ and $L$, the new structure found is shown in Figure.
What shall we get, by joining all the border points of a circular disc?
A quotient space.
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The concept of Nearness
Continuing the gossip, we know neighbourhoods depend on nearness. At this point, let me ask you a question. Who is NEAR to you? My question has embarrassed many; some turning red others turning pale. It is uncivilized to ask personal questions. For some NEAR was his girl friend a thousand kilometers away, to some the boy waiting under the tree, near the bank of the river, to some others the waiting mother at home. NEAR is seldom only taken in distance $-$ sense; no one pointed to the nearest standing / sitting person in reply to my question. Everyone has ones own sense of nearness, varying degrees of nearness in deed. That results in a particular structure for us together.
Let us take an example consisting of myself $a_1$, my wife $a_2$, my elder daughter $a_3$, my younger daughter $a_4$, and neighbours $a_5$, $a_6$. My elder daughter $a_3$ confides everything to me and my wife, but tells us not to divulge certain things to her sister $a_4$. And similar is the response of my daughter $a_4$. Thus closest or smallest neighbourhoods for $a_3$ becomes $\{a_3, a_1, a_2\}$ and for $a_4$ becomes $\{a_4, a_1, a_2\}$. My wife $a_2$, a perfect Indian woman with total commitment to her husband $a_1$, keeps no secret to her without sharing it with me. She loves her daughters much, but surely we the couple do have secrets not to be shared with the daughters. And thus my wife's smallest neighbourhood is $\{a_1, a_2\}$. I do love my wife and children. Fine, but you can't expect me to be as loyal as an Indian wife. Some of my accounts and affairs, I have not shared even with my wife. A self-centred man as I am, my smallest neighbourhood is me myself $\{a_1\}$.
Away from family affairs, let me ask you; ``Have you ever met a saint?'' Let us put the old question to the saint: ``Sir, who is your NEAR one?'' I have received replies: ``My child, who is alien here? All are belonging to my family and are my near ones.'' For a saint, his family is the entire Universe. Those believing otherwise, take it from me, are not genuine saints. In our example $a_5$, $a_6$ are saints; and so even their smallest neighbourhood is $\{a_1, a_2, a_3, a_4, a_5, a_6\}$. The neighbourhood system described above can be represented in the form of a figure.
Neighbourhood system of a family.
Few extra observations to be made:
- $\{a_1, a_2, a_3\}$ together with $\{a_1, a_2, a_4\}$ formulate my home. We may therefore agree to believe that unions of neighbourhoods is a neighbourhood.
- $a_5$ and $a_6$ are FRIENDLY POINTS to any of the congregations, because the largeness of their smallest neighbourhoods embraces each congregation.
- Congregations or collections or sets like $\{a_5, a_6\}$, $\{a_3, a_5, a_6\}$ may be called closed in the sense that they don't have ``FRIENDLY POINTS'' outside themselves.
- Congregations like $\{a_1, a_2, a_3\}$, $\{a_1, a_2, a_4\}$ are characterized by the fact that no member or point in it is a friendly point of outsiders. Compare $(3)$ and $(4)$.
The individuals $a_1, a_2, a_3, a_4, a_5, a_6$, if all were like
saints, neighbourhood for any of the points would be $\{a_1, \,
\ldots,\, a_6 \}$; and the structure would be represented as in
figure.
Neighbourhood system when every member is a saint.
And if each of $a_1, \, \ldots,\, a_6$ was to be as self-centred as myself, God forbid, then each $\{a_1\}$, $\{a_2\}$, $\{a_3\}$, $\{a_4\}$, $\{a_5\}$, $\{a_6\}$ would be neighbourhoods of $a_1, \, \ldots,\, a_6$, respectively. It is indeed fortunate that in the real world, only few are selfish like me. Remember it would be equally horrible, if all were to be saints.
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A closer look at the reals
Every branch of Mathematics depends on the real numbers, we use real numbers to simplify things and also to complicate it because the real number system itself is deceptive and slippery to handle. The real analysis influenced the subject of the study of neighbourhoods so much resulting in the development of Point set topology that Poincare, the father of the subject termed the new development as a ``disease'' to be overcome by future Mathematicians. Poincare's prophesied ``Future'' has been unfortunately captured by the ``disease''. For a look at the real numbers we may look back to the days, when we were learning counting: one, two, three, $\ldots$, hundred, $\ldots$ thousand, ..., million, ..., trillion. It ends not if we counted all the lifelong. This unendingness does thrill everyone leading to school days fantasies like: ``I have more than what you can think of''. It was a wonder that in a mortal world, there was something not to end. Then in early college days we come across things like a collection of all positive real numbers which are less than one, usually denoted by $(0,1)$. Here again comes something ``without an end'', without a last number. The surprise is greater because the unendingness does not require the monstrosity of largeness-unabated of the earlier case. The unendingness (its Sanskrit equivalent ``Anant'') was invariably associated with trans-finite. And look! The set $(0, 1)$, very much in finite world has no end. It is remarkable that $(0, 1)$ has no beginning (``Anadi'' in Sanskrit) either. In Hindu religion ``Anadi'' and ``Anant'' are the characteristics attributed to God (Brahma), the only God. Here emerges another .... !! Again, look at the representation $(0, 1)$ of the set chosen. What you see with eyes is the zero and the one, in the representation. Does zero belong to the set $(0, 1)$? No. Does one belong to the set $(0, 1)$? Again no. And the representation figures zero and one only. The numbers $1/2, 1/3, 5/6$ ... etc ... belonging to the set don't find a place in the representation. That is to say, the members belonging to the set, we don't see in the representation, and what we see in the representation does not belong to the set. Amusing! To see God, we close our eyes. Now, to see the set $(0, 1)$, better close your eyes. With closed eyes, you may see Ghosts as well, courtesy `Real Numbers'. It is easily observed that the open intervals $(3,5)$, $(-4,20)$, $(-100,100)$, ..., $(- \infty, \infty)$ are structurally (topologically) similar (homeomorphic) to $(0,1)$. Any sample is good for study, let us explore the Brahma Tatva (the atomic power) hidden in an interval. We ask, which of the sets below shall we accept as neighbourhoods of the point $0$ in the real line? $(i) \{0, 1, 2, 3 \}$ $(ii) [-1, 1]$ $(iii) \{-3, -2, -1, 0, 1, 2, 3\}$ $(iv) (-1, 1)$ $(v) [-5, 5] \cup \{7, 8, 9\}$ $(vi) [-10, 10] \cup \{11, 12, 13, ..... \}$ $(vii) (-1/100, 1/10)$ $(viii) [-2,\, \infty )$ $(ix) [0, 5]$. Nearby (nearness in the sense of distance in $\mathbb{R}$) points $1/2, 1/5, 1/100,\, \ldots $ are ignored in $(i)$. It is indeed disrespectful to all the points in the internal $(0,1)$, which is nearer to zero than are the points $1, 2$ or $3$ of $(i)$. So $(i)$ deserves to be rejected and $(iii)$ also deserved to be rejected on similar grounds. We may also observe that were $(i)$ and $(iii)$ to be accepted as neighbourhood, then even the set $\{0\}$ will have to be accepted as a neighbourhood of $0$. That was surely not the intuitive idea of the neighbourhood. The $(ii)$ seems acceptable as neighbourhood of $0$, so also the set $(-1, 1)$ in $(iv)$. It occurs here that the set $[-1, 1]$ is a neighbourhood as it is something extra (containing) than the neighbourhood $(-1, 1)$. The set $(vii)$, as also the set $(iv)$ and $(viii)$ contains an entire interval on the right side as well as on the left side, not necessarily equal on the two sides, and thus seem acceptable as neighbourhoods. While inviting for a party, it is natural to have neighbours both from the left as well as from the right of your home, not necessarily the same number from both sides (believing there is no neighbours in the front on back, above or below, to confine to one-dimensional $\mathbb R$). The set in $(ix)$ has the shortcoming of not having left sided neighbourhood. The common feature in the acceptable sets to be neighbourhood emerges out to be containing some length of interval on both sides of the point. In the examples here $(vii)$ is a neighbourhood of $0$ implies $(ii)$, $(iv)$, $(v)$, $(vi)$ and $(viii)$ to be also neighbourhoods of $0$. Something very special about the sets $(-1, 1)$ and $(-1/100, 1/10)$ in the above is that they are not only neighbourhood of the point $0$ but are neighbourhood of each point belonging to the set. Open intervals cannot be trivial and closed intervals like $[0, 1]$ are non-trivial because they contain non-trivial sets like open interval $(0, 1)$, which are neighbourhood of each of its points. Closed interval $[0, 1]$, though not trivial, lacking the non-beginning and non-ending property lacks the wild and explosive strength of the open interval $(0,1)$. Addition of points $0$ and $1$ to $(0, 1)$ is indeed taming the wild set $(0, 1)$, while addition of points $5, \, 6, \, 7, \, 8, \, \ldots\, $ would not tame it. The tamed set $[0, 1]$ behaves like a finite set (see compactness). The non-beginning and non-ending properties of the set $(0, 1)$ are indeed providing it infinite character at terminals (better to be called non-terminals). This provision in turn provides for the possibility of unbondedness of continuous functions or non-uniformness of continuity. Try to make; actually drawing graphs of continuous functions, a function unbounded on the closed interval $[0, 1]$ or $[4, 5]$. And realise the need of a non-beginning or non-ending place in the domain to succeed in your effort. See, for instance, figure.
Unboundedness, non-beginningness and non-endness.
You must observe how addition of points $0$, $1$ to the set $(0, 1)$ is a taming operation, while addition of {$5, 6, 7, 8, 9$} to $(0, 1)$ is not one such. \vspace*{1mm} Observe the difference that neighbourhoods make. This unwanted digression was caused due to familiarity with real analysis. What we had actually found special about open intervals: ``being neighbourhood of each of its points'', is also true for the unions of open intervals called ``open sets''. The discussion above, putting it the other way round, can be summed up as follows: ``once the open sets are known, neighbourhood of each of the points can be determined'', and so the structure of the object can be determined. For this reason, to describe the structure of an object, a declaration of open sets called ``topology'' alone is made. For example the neighbourhood structure in example of the article "The Concept of Nearness" is evident from the topology
\[ \begin{array}{l} \{ \{a_1\},\, \{ a_1,\, a_2\},\, \{ a_1,\, a_2,\, a_3\},\, \{ a_1,\, a_2,\, a_4\},\\ \qquad \qquad \qquad \qquad \, \{ a_1, a_2, a_3, a_4\},\, \{ a_1,\, a_2,\, a_3,\, a_4,\, a_5,\, a_6\},\, \phi \}. \end{array} \]
The inclusion of the empty set $\phi$ is a courtesy for convenience. Our knowledge of the collection of open sets in our familiar cases guides us to impose a discipline in the declaration of topology:
- Unions of declared sets be included in the declaration.
- Finite intersection of declared sets be included in the declaration.
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Understanding the newborn
We started the preceding section by observing some slippery properties of real numbers. Is it not amusing that what comes next to zero on the number line, we don't know. What precedes zero, which too we don't know. What real number comes next to five, we don't know. Take any real number, and we have ignorance immediately following it and immediately preceding it. This ignorance is a corollary to the axiom that between any two real numbers there lies another real number and the axiom of no gap in the reals is of tremendous value for the study of topology of the reals. From the axiom of overcrowding, it follows that a point of the real line does not have a smallest open set (or neighbourhood) unlike in an example of Section IV. We can see that \[ (-1, 1),\ (-1/2, 1/2) ,\, \ldots,\, (-1/20, 1/20),\ \ldots \] are smaller and smaller open sets containing zero (or neighbourhood of zero); and any other neighbourhood of $0$ must contain a member of the system of neighbourhoods just described. (More on reals elsewhere, away from the study of neighbourhoods).
Customarily, bridging algebra of reals to geometry, the set of all real numbers is represented as a straight line, without a beginning and without an end; marked with $\ldots,\, -3,\, -2,\, -1,\, 0,\, 1,\, 2,\, 3,\, 4,\, 5,\, \ldots$ at unit intervals, as shown in figure.
The Real Line.
To accommodate more of the real line on our paper, we may turn it as if it was a rubber-pipe with markings.
Twisted Rubber Pipe
We remember that the turnings do not change the topological character of the rubber pipe (the number line). Suppose our rubber pipe develops few cracks here and there between $(-1, 10]$.
Cracked portion $(-1, 10]$ removed.
To make the remaining pipe usable, we cut out the cracked leaking portion from $(-1, 10]$ (see Figure \cracked portion...) and join the remaining parts. The rejoined pipe or the repaired line perform as nicely as the original one did, and they seem to be structurally the same as before.
Left over parts $(-\infty, -1]$ and $(10, \infty)$ joined.
We must, however, pause to observe the markings on the repaired line. We now find that $-1$ is immediately followed by $(10, 10 + p)$, where $p > 0$; and preceded by $(-1-q, -1)$, where $q > 0$. Thus neighbourhoods of $-1$, would be of the form $(-1 - q, -1]\, \cup\, (10, 10 + p)$, $p > 0$, $q > 0$. Neighbourhoods of other left over points is easily available as that part of the original neighbourhoods that occurs in $(-\infty, -1] \,\cup\, (10, \infty)$. And this repaired line is topologically same as the original real line. I am sure, you will not use the repaired line for arithmetical addition etc. If the removed portion was $[-1, 10)$, you know the method of analogous surgery.
Let us consider the case: the removed portion being of the form $[-1, 10]$. The natural question is whether the joined $(-\infty, -1) \, \cup \, (10, \infty)$ is even now structurally the same as the original line? And if it is not, then what amendments would make it behave and perform as the original line or the original pipe? The intuition suggests that a smooth flow through $(-\infty, -1)$ into $(10, \infty)$ is not possible,by merely readjusting the neighbourhoods of the left over points. The failure lies in the absence of a point in the left over portions that may, with suitable neighbourhoods, perform the joining. So a bridging single point (or a bridge length) is to be provided for the joining. By adjoining the point $10$, with suitable neighbourhoods or else adjoining the point with suitable neighbourhoods would perform the joining.
Can't we join the two parts by something else than $-1$ or $10$? Can't we try $7$ to perform the glueing job? That is to say, we want $7$ to perform the act of hooking up the two parts $(- \infty, -1 )$ and $( 10, \infty )$. What is this hooking up act of $7$? This action is achievable by defining the neighbourhoods of $7$ as $(-q -1, -1) \cup \{7\} \cup (10, 10 + p)$, $p > 0$, $q > 0$. Proximity with a set; of a point is thus achievable by the instrument of neighbourhoods. The role of $7$ could be performed by $0,\, 2,\, 4,\, 7,\, \sqrt{137}$, etc with analogous neighbourhoods defined for them. Here $7$, $0$, $2$, $4$, .... etc are able to perform their bridging role due to the neighbourhoods assigned . Not to perform any arithmetic $7$, $0$, $2$, $4$, ... etc has no numerical significance. What $7$ does or $2$, $4$, ... etc can do; can be achieved by any symbol say $\alpha$ with appropriate neighbourhoods. That is to say with neighbourhoods of the form $(- q - 1 , - 1) \cup \{\alpha\} \cup (10, 10+p)$, $p > 0 $, $q > 0$ provides a structure to $(-\infty, -1) \cup \{\alpha\} \cup (10,\infty)$ which is structurally the same as that of the real line $(-\infty , \infty)$.
How different neighbourhood systems assigned to, provide topologically different structures is shown in following figures.
Neighbourhoods of $\alpha$ of the form $( -1 - q,\, -1) \cup \{\alpha\} \cup (10, 10+p)$; $\alpha$ hooks up the two parts and $(-\infty,\, -1) \cup \{\alpha \} \cup (10, \infty)$ is structurally same as $(-\infty,\, \infty)$.
Neighbourhoods of $\alpha$ of the form $\{\alpha\} \cup (10,10+p)$; the same set$ (-\infty, \, -1) \cup \{\alpha \} \cup (10, \infty)$ is structurally same as $(-\infty, -1) \cup [10,\infty)$, a two pieced structure.
Neighbourhoods of $\alpha$ the form $(-1-q, -1) \cup \{\alpha \}$; the same set $(-\infty, -1)\cup \{\alpha\} \cup (10, \, \infty)$ becomes structurally same as $(-\infty, -1] \cup (10, \infty)$.}\label{left-side
$\{\alpha\}$ is a neighbourhood of $\alpha$, and now the same set $(-\infty,\, -1) \cup \{ \alpha \} \cup (10,\, \infty)$ is structurally same as $(-\infty, \, -1) \cup \{7\} \cup (10, \infty)$; a model with three pieces.
One point more $-$ in the above discussion we have seen that a harmonious joining of the two parts is possible only if the parts are of the form $( -\infty, 1]$ and $( 10, \infty)$ or of the form $( -\infty, 1)$ and $[10, \infty)$. We may agree to call sets of the form $(-\infty, 1]$ or $[10, \infty)$ as male, while the sets of the form $( -\infty, 1 )$ or $( 10, \infty )$ as female. Our conclusion, in other words means that harmonious blending into a continuum is possible only if of the two constituent sets one is male and the other female.
The converse $-$ a physical problem puzzles me. It says that if an endless thread is to be broken into two connected pieces then one must be male and another female? Splitting of objects of dimension higher shall have similar conclusions. Before exploring that, firstly take a piece of thread, break it into two and determine the sex of the two parts !!!
We may not forget that females like $( -\infty, 1)$ or $(10, \infty)$ are structurally, even without joining a male are already the same as the continuum $(-\infty, \infty)$, endowed with the power of Brahma, referred to else where.
As our last example, for the present discussion; consider the set of non-negative real numbers less than $1$, denoted as $[0, 1)$. Let the neighbourhoods of $0$ be taken as sets of the form $[0, p) \cup (1-q, 1)$, $p > 0$, $q > 0$. For other points, neighbourhoods being taken to be the part of neighbourhoods in $\mathbb{R}$ occurring in $[0, 1)$. The dead-endness of the set at $0$ and endlessness before $1$ is seen to be replaced by a continuous flow: as though the sequence of points $95/100$, $96/100,\ \ldots,\, 97/100$, $\ldots,\, 9999/10000$, $\ldots,\, 0,\, \ldots$, $1/10000,\, \ldots$, $ 1/1000$, $\ldots,\, 1/100$, $\ldots,\, 2/100$, $\ldots,\, 3/100,\, \ldots,\, 95/100,\, \ldots$ follow a circular path as in following figure.
New structure of $(0, 1]$.
It indeed defines a cyclic structure $-$ no ends. Anyone finding the phenomenon unnatural and not acceptable to their taste might look at a world map, representing the globe. Take a point in the east on the map, move eastwards, and go on ....... till you emerge from the far-west.
We have also described a cyclic path, reaching discussions similar to made under the heading "New Structures".
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Some further discussions
For a smooth discussion of the subject, few more basic concepts be understood.
Discussion A [Friendly Points]
Consider the following sets of reals:
- $\{ 1/2,\, 2/3,\, 3/4,\, \ldots,\, 99/100,\, \ldots,\, 9999/10000,\, \ldots \}$
- $[0, 1]$
- $[0, 1)$
- $ (1,100] $
- $[0, 5]$
- $ [1/2, 9/10] \cup \{ 0, \, 99/100, \, 999/1000, \, 1, \, 5, \, 7 \}$
- $\{ 0,\, 99/100,\, 3,\, 15,\, 55,\, 500\}$.
We thus find that if a point is friendly to a set then each neighbourhood of the point would have a point of the set. So that each point of the set; even aloof and isolated ones belonging to the set; are not to be counted as friendly, we insist that each neighbourhood of a friendly point should have a point of the set other than the point itself.
Thus vaguely speaking, the friendly points to a set or a family are characterized by the fact that in each party hosted by `a friend' some part of the family is always represented. That is to say that each neighbourhood of `a friend' has some member of the set. Points of the set accumulate or crowd about a friendly point.
Let us again look at the set $(6)$ and the status of the point $1$ in it. In the set $(6)$, that is, $[1/2, 9/10] \cup \{ 0,\, 99/100,\, 999/1000,\, 1,\, 5,\, 7\}$, the point $1$ does not seem to be friendly to the set. The point $1$ seems more like someone sitting away from the group, eating his tiffin alone. Such aloof and `isolated' points suffer from problems like `Ego' or `Infectious Disease'. To
discriminate these isolated points of the set from insider friendly points of a set, we must revise our characterization of `friendly' points.
A point be called `friendly' to a set if each of its neighbourhood contains a member of the set other than itself also. The friendly points are termed `limit points' and sets not having friendly fellows outside itself are termed closed. It is easy and useful to observe that a set in a topological space is closed if and only if its complement is open.
Note that the friendly points of a set can be an `outsider' as well as an `insider'. For the set $(0, 1)$ in $\mathbb{R}$, each point of the set is `friendly' to the set; and outsiders $0$ and $1$ are also friendly to the set.
Friendly points in some cases have even more to do. Let us observe how, in some cases, they are instrumental in ``joining'' different sets.
The two disjoint sets $(- \infty,\, 1)$ and $[-1, \infty)$ together make a continuum. The point $1$ performs the joining act. In a previous article title ``Understanding the Newborn'', we found that an abstract $\alpha$, with its neighbourhoods of the form $(- q - 1, \, -1) \cup \{ \alpha \} \cup (10, 10+p)$, $p > 0$, $q > 0$ performs the joining act between $(-\infty, \, -1)$ and $(10, \infty)$. It is the friendliness of $1$ to the sets $(-\infty,\, 1)$ and $[1,\, \infty )$, and in the earlier example that of $\alpha$ to the sets $(-\infty,\, -1)$ and $(10, \infty)$ that performs the joining act.
Discussion B [Distance Structured Spaces]
It may be observed that much of Geometry and Analysis of the real line $\mathbb{R}$, the Euclidean spaces $\mathbb{R}^2$, $\mathbb{R}^3$, follow from the distance formula (Pythagorean metric) and the open sets (or topology) it determines. It is, therefore natural to study the metric spaces as special topological spaces. And then among metric spaces too, more interesting are those, where the underlying set has an algebraic structure (vector-spaces; allowing addition and scalar multiplication of its members).
Now studying topological spaces, we keep looking for those ones, which though not metric spaces, satisfy some basic properties true in metric space; thus behaving somewhat like metric spaces. An example: spaces in which any two points have disjoint neighbourhoods are called Hausdorff spaces. Topological spaces with these special properties have been found very useful; and have been extensively studied.
Discussion C [A Foundation for the Open Sets]
To specify a topological space, quite often, it is found unnecessary to describe or mention all its open sets. What we might do is to mention some more basic ones, and expect that the others are understood by taking their unions. For example for the real line (with usual topology induced by the distance), a mention of all the open intervals is sufficient, the open sets of real are only their unions. And in case of metric space, a mention of open spheres - suggests the topology. This part-topology is called a base for the topology. Similarly, yet a part of the base is called a subbase for the topology, if the family of its finite intersections is a base for the topology. For instance, the collection of infinite intervals of the form $( - \infty, \, b), \ (a, \, \infty )$, $a, b \in \mathbb{R}$ is easily seen to be a subbase for the usual topology on $\mathbb{R}$.
For description of the product space, we might mention here that the concept of base is seen to be very useful.
No study of topology is possible and meaningful without studying the concept of homeomorphisms and continuity of mappings $-$ the core of analysis. To give the concept its due respect, we discuss it separately.
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