For this purpose, we introduce a natural topology on Milnor's K-groups [K.sup.M.sub.l](k) for a topological field k as the quotient topology induced by the joint determinant map and show that, in case of k = R or C, the natural topology on [K.sup.M.sub.l](k) is disjoint union of two indiscrete components or indiscrete topology, respectively. Some "extremal" examples Take any set X and let = {, X}. 7. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. R := R R (cartesian product). [Justify your claims.] Is the subspace topology of a subset S Xnecessarily the indiscrete topology on S? Let Xbe a topological space with the indiscrete topology. R under addition, and R or C under multiplication are topological groups. The indiscrete topology on Xis de ned by taking Ëto be the collection consisting of only the empty set and X. The same argument shows that the lower limit topology is not ner than K-topology. SOME BASIC NOTIONS IN TOPOLOGY It is easy to see that the discrete and indiscrete topologies satisfy the re-quirements of a topology. If X is finite and has n elements then power set of X has _____ elements. In the discrete topology, one point sets are open. Then, clearly A\B= ;, but A\B= R 0 \R 0 = f0g. To learn more, see our tips on writing great answers. Let Ï be the collection all open sets on X. The largest topology contains all subsets as open sets, and is called the discrete topology. If Adoes not contain 7, then the subspace topology on Ais discrete. [Justify your claims.] TOPOLOGY TAKE-HOME CLAY SHONKWILER 1. Is there a difference between a tie-breaker and a regular vote? When k = R and l [greater than or equal to] 2, G either is an indiscrete space or has an indiscrete subgroup of index 2. This agrees with the usual notation for Rn. Removing just one element of the cover breaks the cover. $(0,1)$ is compact in discrete topology on $\mathbb R$. C The lower-limit topology (recall R with this the topology is denoted Râ). Why set of integer under indiscrete topology is compact? In parliamentary democracy, how do Ministers compensate for their potential lack of relevant experience to run their own ministry? Conclude that if T ind is the indiscrete topology on X with corresponding space Xind, the identity function 1 X: X 1!Xind is continuous for any topology T 1. 7. Let Bbe the collection of cartesian product of open intervals, (a;b) (c;d). (viii)Every Hausdorspace is metrizable. Let X be any set and let be the set of all subsets of X. contains) the other. The sets in the topology T for a set S are defined as open. valid topology, called the indiscrete topology. valid topology, called the indiscrete topology. Ø®ÓkqÂ\O¦K0¤¹@B Let V ï¬ zPU B 1 7 pzq. Then is a topology called the trivial topology or indiscrete topology. (b) Any function f : X â Y is continuous. 3. If we use the discrete topology, then every set is open, so every set is closed. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. In the indiscrete topology all points are limit points of any subset X of S which inclues points other than because the only open set containing a point p is the whole S which necessarily contains points of ⦠How/where can I find replacements for these 'wheel bearing caps'? In the discrete topology - the maximal topology that is in some sense the opposite of the indiscrete/trivial topology - one-point sets are closed, as well as open ("clopen"). because it closed and bounded. (Lower limit topology of R) Consider the collection Bof subsets in R: B:= Page 1. C The lower-limit topology (recall R with this the topology is denoted Râ). Proof. When \(\mathcal{T} = \{\emptyset, X\}\), it is called the indiscrete topology on X. 5. , the indiscrete topology or the trivial topology on any set X. Use MathJax to format equations. Then Z = {α} is compact (by (3.2a)) but it is not closed. Then, clearly A\B= ;, but A\B= R 0 \R 0 = f0g. c.Let X= R, with the standard topology, A= R <0 and B= R >0. TOPOLOGY TAKE-HOME CLAY SHONKWILER 1. As for the indiscrete topology, every set is compact because there is only one possible open cover, namely the space itself. Proposition. Intersection of Topologies. The indiscrete topology is manifestly not Hausdorâµunless X is a singleton. Sierpinski Space 44 12. Is there any source that describes Wall Street quotation conventions for fixed income securities (e.g. There are also infinite number of indiscrete spaces. Confusion about definition of category using directed graph. Since that cover is finite already, every set is compact. (b) Suppose that Xis a topological space with the indiscrete topology. My professor skipped me on christmas bonus payment, How to gzip 100 GB files faster with high compression. Is it true that an estimator will always asymptotically be consistent if it is biased in finite samples? A The usual (i.e. Indiscrete topology is finer than any other topology defined on the same non empty set. The Discrete Topology Let Y = {0,1} have the discrete topology. corporate bonds)? 2. Why? Are they homeomorphic? Some "extremal" examples Take any set X and let = {, X}. A Topology on Milnor's Group of a Topological Field ⦠Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Example: The indiscrete topology on X is Ï I = {â
, X}. Let $X=\mathbb R$ with cofinite topology and $A=[0,1]$ with subspace topology - show $A$ is compact. Then Z is closed. and x 1. with the indiscrete topology. 38. Let X be the set of points in the plane shown in Fig. Ú Thanks for contributing an answer to Mathematics Stack Exchange! Here are four topologies on the set R. For each pair of topologies, determine whether one is a reï¬nement of (i.e. but the same set is not compact in indiscrete topology on $\mathbb R$ because it is not closed (because in indiscrete topolgy on $\mathbb R$ the closed sets is only $\phi$ and $\mathbb R$). Indiscrete Topology The collection of the non empty set and the set X itself is always a topology on X,⦠Click here to read more. Proof. Partition Topology 43 6. standard) topology. However: (3.2d) Suppose X is a Hausdorï¬ topological space and that Z â X is a compact sub-space. (Lower limit topology of R) Consider the collection Bof subsets in R: B:= The standard topology on R induces the discrete topology on Z. Consider the set X=R with T x = the standard topology, let f be a function from X to the set Y=R, where f(x)=5, then the topology on Y induced by f and T x is. For this purpose, we introduce a natural topology on Milnor's K-groups [K.sup.M.sub.l](k) for a topological field k as the quotient topology induced by the joint determinant map and show that, in case of k = R or C, the natural topology on [K.sup.M.sub.l](k) is disjoint union of two indiscrete components or indiscrete topology, respectively. The properties verified earlier show that is a topology. Let Bbe the collection of cartesian product of open intervals, (a;b) (c;d). Ë is a topology on . So you can take the cover by those sets. A The usual (i.e. with the indiscrete topology. c.Let X= R, with the standard topology, A= R <0 and B= R >0. Such spaces are commonly called indiscrete, anti-discrete, or codiscrete. Then Xis compact. If Adoes not contain 7, then the subspace topology on Ais discrete. These sets all have in nite complement. space. The dictionary order topology on the set R R is the same as the product topology R d R, where R d denotes R in the discrete topology. Closed Extension Topology 44 13. By clicking âPost Your Answerâ, you agree to our terms of service, privacy policy and cookie policy. Proof We will show that C (Z). Page 1. If Xhas at least two points x 1 6= x 2, there can be no metric on Xthat gives rise to this topology. The standard topology on R n is Hausdorâµ: for x 6= y 2 R n ,letd be half the Euclidean distance ⦠X with the indiscrete topology is called an. The collection of the non empty set and the set X itself is always a topology on X, and is called the indiscrete topology on X. 2) ËË , ( Ë Ë power set of is a topology on and is called discrete topology on and the T-space Ë is called discrete topological space. Let X be any set and let be the set of all subsets of X. indiscrete). Where can I travel to receive a COVID vaccine as a tourist? Here, every sequence (yes, every sequence) converges to every point in the space. Example 2.4. The indiscrete topology on Xis de ned by taking Ëto be the collection consisting of only the empty set and X. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Finite Excluded Point Topology 47 14. is $(0,1)$ compact in indiscrete topology and discrete topolgy on $\mathbb R$, Intuition for the Discrete$\dashv$Forgetful$\dashv$Indiscrete Adjunction in $\mathsf{Top}$, $\mathbb{Q}$ with topology from $\mathbb{R}$ is not locally compact, but all discrete spaces are, Intuition behind a Discrete and In-discrete Topology and Topologies in between, Fixed points Property in discrete and indiscrete space. 10/3/20 5: 03. Making statements based on opinion; back them up with references or personal experience. Expert Answer 100% (1 rating) Previous question Next question Get more help from Chegg. 2) ËË , ( Ë Ë power set of is a topology on and is called discrete topology on and the T-space Ë is called discrete topological space. In the discrete topology - the maximal topology that is in some sense the opposite of the indiscrete/trivial topology - one-point sets are closed, as well as open ("clopen"). If Xhas at least two points x 1 6= x 2, there can be no metric on Xthat gives rise to this topology. Expert Answer . Compact being the same as closed and bounded only works when $\mathbb{R}$ has the standard topology. So the equality fails. Let S Xand let T S be the subspace topology on S. Prove that if Sis an open subset of X, and if U2T S, then U2T. Let X be the set of points in the plane shown in Fig. That's because the topology is defined by every one-point set being open, and every one-point set is the complement of the union of all the other points. Previous question Next question Transcribed Image Text from this Question. If one considers on R the indiscrete topology in which the only open sets are the empty set and R itself, then int([0;1]) is the empty set. There are also infinite number of indiscrete spaces. 1.1.4 Proposition (a) Let Xbe a set with the co nite topology. 10/3/20 5: 03. It also converges to 7, e, 1;000;000, and every other real number. 1is just the indiscrete topology.) 3.Let (R;T 7) be the reals with the particular point topology at 7. The Discrete Topology Let Y = {0,1} have the discrete topology. ±.&£ïBvÙÚg¦m ûèÕùÜËò¤®Õþ±d«*üë6þ7Í£$D`L»ÏÊêqbNÀ÷y°¡Èë$^'ÒBË¢K`ÊãRN$¤à½ôZð#{øEWùz]b2Áý@jíÍdº£à1v¾Ä$`ÇáæáwÆ Then is a topology called the trivial topology or indiscrete topology. (In addition to X and we ⦠Proof. The same argument shows that the lower limit topology is not ner than K-topology. In (R;T indiscrete), the sequence 7;7;7;7;7;::: converges to Ë. Let R 2be the set of all ordered pairs of real numbers, i.e. 6. , the finite complement topology on any set X. Proposition 18. Similarly, if Xdisc is the set X equipped with the discrete topology, then the identity map 1 X: Xdisc!X 1 is continuous. 'øÈ÷¡àItþ:N#_ÉÂ#1NÄ]¤¸¬ F8µ$üù â¥n*q/öúyæMR«î«öjR(@Ï:,½PýT©mªUlºÆ¢Ã}Ø1Öé13&ô9ÐÁeQnÉ@ñß] 6J l¤ôÔ~¸KÚ¢ "çQ"ÔÈq#/C°Y0. This topology is called indiscrete topology on and the T-space Ë is called indiscrete topological space. In this, we use a set of axioms to prove propositions and theorems. X with the indiscrete topology is called an. 3.Let (R;T 7) be the reals with the particular point topology at 7. So the equality fails. space. 2.13.6. Let f : X !Y be the identity map on R. Then f is continuous and X has the discrete topology, but f(X) = R does not. but the same set is not compact in indiscrete topology on R because it is not closed (because in indiscrete topolgy on R the closed sets is only Ï and R). Don't one-time recovery codes for 2FA introduce a backdoor? 1.A product of discrete spaces is discrete, and a product of indiscrete spaces is indiscrete. If we use the indiscrete topology, then only â
,Rare open, so only â
,Rare closed and this implies that A ⦠Similarly, if Xdisc is the set X equipped with the discrete topology, then the identity map 1 X: Xdisc!X 1 is continuous. Then \(\tau\) is called the indiscrete topology and \((X, \tau)\) is said to be an indiscrete space. Note that there is no neighbourhood of 0 in the usual topology which is contained in ( 1;1) nK2B 1:This shows that the usual topology is not ner than K-topology. The dictionary order topology on the set R R is the same as the product topology R d R, where R d denotes R in the discrete topology. Proposition. On the other hand, the union S x6=x 0 fxgequals Xf x 0g, which has complement fx 0g, so it is not open. In fact no infinite set in the discrete topology is compact. \(2^n\) \(2^{n-1}\) \(2^{n+1}\) None of the given; The set of _____ of R (Real line) forms a topology called usual topology. 21 November 2019 Math 490: Worksheet #16 Jenny Wilson In-class Exercises 1. It su ces to show for all U PPpZq, there exists an open set V â¢R such that U Z XV, since the induced topology must be coarser than PpZq. If we thought for a moment we had such a metric d, we can take r= d(x 1;x 2)=2 and get an open ball B(x 1;r) in Xthat contains x 1 but not x 2. We are only allowing the bare minimum of sets, X and , to be open. ) (a) X has the discrete topology. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Zvi Rosen Applied Algebraic Topology Notes Vladimir Itskov 3.1. Review. (R Sorgenfrey)2 is an interesting space. Here are four topologies on the set R. For each pair of topologies, determine whether one is a reï¬nement of (i.e. This topology is called indiscrete topology on and the T-space Ë is called indiscrete topological space. 2. indiscrete topology 3. the subspace topology induced by (R, Euclidean) 4. the subspace topology induced by (R, Sorgenfrey) 5. the finite-closed topology 6. the order topology. 2 CHAPTER 1. This implies that A = A. Proof. Then Xis not compact. If A R contains 7, then the subspace topology on Ais also the particular point topology on A. This is the space generated by the basis of rectangles Chapter 2 Topology 2.1 Introduction Several areas of research in modern mathematics have developed as a result of interaction between two or more specialized areas. Thus openness is not a property determinable from the set itself; openness is a property of a set with respect to a topology. If we thought for a moment we had such a metric d, we can take r= d(x 1;x 2)=2 and get an open ball B(x 1;r) in Xthat contains x 1 but not x 2. X with the indiscrete topology is called an indiscrete topological space or simply an indiscrete space. Proposition 17. Note that there is no neighbourhood of 0 in the usual topology which is contained in ( 1;1) nK2B 1:This shows that the usual topology is not ner than K-topology. Let X = R with the discrete topology and Y = R with the indiscrete topol- ogy. 4. In the discrete topology any subset of S is open. 38. This question hasn't been answered yet Ask an expert. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Uncountable Particular Point Topology 44 11. (a) Let (X;T) be a topological space. rev 2020.12.10.38158, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Terminology: gis the genus of the surface = maximal number of ⦠Subscribe to this blog. is $(0,1)$ compact in indiscrete topology and discrete topolgy on $\mathbb R$? \(2^n\) \(2^{n-1}\) \(2^{n+1}\) None of the given; The set of _____ of R (Real line) forms a topology called usual topology. (R usual)2 = R2 usual. Usual Topology on $${\mathbb{R}^3}$$ Consider the Cartesian plane $${\mathbb{R}^3}$$, then the collection of subsets of $${\mathbb{R}^3}$$ which can be expressed as a union of open spheres or open cubed with edges parallel to coordinate axis from a topology, and is called a usual topology on $${\mathbb{R}^3}$$. !Nñ§UD
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?§ÂQ4X¦Taðå«%x¸!iT 4K. 6. Example 2. (c) Any function g : X â Z, where Z is some topological space, is continuous. The indiscrete topology on X is the weakest topology, so it has the most compact sets. How to holster the weapon in Cyberpunk 2077? This is the next part in our ongoing story of the indiscrete topology being awful. Is it just me or when driving down the pits, the pit wall will always be on the left? Then \(\tau\) is called the indiscrete topology and \((X, \tau)\) is said to be an indiscrete space. (This is the opposite extreme from the discrete topology. Then Z = {α} is compact (by (3.2a)) but it is not closed. For example, t Example 1.5. Indiscrete Topology 42 5. When should 'a' and 'an' be written in a list containing both? In particular, every point in X is an open set in the discrete topology. The indiscrete topology for S is the collection consisting of only the whole set S and the null set â
. How do I convert Arduino to an ATmega328P-based project? contains) the other. Every sequence converges in (X, Ï I) to every point of X. Let Xbe a topological space with the indiscrete topology. In other words, for any non empty set X, the collection $$\tau = \left\{ {\phi ,X} \right\}$$ is an indiscrete topology on X, and the space $$\left( {X,\tau } \right)$$ is called the indiscrete topological space or simply an indiscrete space. It is called the indiscrete topology or trivial topology. The is a topology called the discrete topology. 1. The intersection of any two topologies on a non empty set is always topology on that set, while the union⦠Click here to read more. Proof. TSLint extension throwing errors in my Angular application running in Visual Studio Code. Asking for help, clarification, or responding to other answers. [note: So you have 4 2 = 6 comparisons to make.] of X X X, and so on. Let X be the set of points in the plane shown in Fig. We sometimes write cl(A) for A. The is a topology called the discrete topology. (c) Any function g : X â Z, where Z is some topological space, is continuous. R := R R (cartesian product). (iii) The cofinite topology is strictly stronger than the indiscrete topology (unless card(X) < 2), but the cofinite topology also makes every subset of ⦠1.1.4 Proposition (Limits of sequences are not unique.) Then Xis compact. 2 CHAPTER 1. [note: So you have 4 2 = 6 comparisons to make.] Let Xbe an in nite topological space with the discrete topology. Balls in metric c.Let X= R, with the standard topology on S question Next Get... { R } $ has the most compact sets in fact, with the and. Or when driving down the pits, the pit wall will always asymptotically be consistent if it is ner! 1 ; 000 ; 000 ; 000, and every other real number is closed topology trivial... By ( 3.2a ) ) but it is easy to see that the topologies of R the... Let ( X ; T ) be the set of integer under topology. Service, privacy policy and cookie policy when $ \mathbb R $ not closed called! Consisting of only the whole set S and the topology is called indiscrete topology indiscrete. Mis orientable, M= H ( g ) = # g 2 Previous. Back them up with references or personal experience, anti-discrete, or codiscrete do I convert Arduino to an project. Compensate for their potential lack of relevant experience to run their own ministry is easy see! One-Time recovery codes for 2FA introduce a backdoor of R2 it true that an estimator will asymptotically. That Z â X is a topology the co nite topology, ( a ) let ( X and! Like other branches of pure mathematics, is an interesting space question has n't answered. 0,1 ) $ compact in indiscrete topology is denoted Râ ) earlier show that c ( Z ) contains. For contributing an indiscrete topology on r to mathematics Stack Exchange is a Hausdorï¬ topological,... V is open since it is easy to see that the lower limit topology is than! ' be written in a list containing both 3 de 12. indiscrete topological space X the following are.. As for the indiscrete topol- ogy topology generated by Bis called the trivial topology on Xis de by! Of X has _____ elements with references or personal experience topology defined the. And professionals in related fields like other branches of pure mathematics, is continuous is denoted Râ ) or. Any subset of S is open. any other topology defined on the set of points in the topology! And, to be open. sequence ) converges to every point indiscrete topology on r! And, to be open. in ( X, and is called indiscrete topology for S is.! M= M ( g ) = # gRP2 simply an indiscrete, Ï I ) to every of... Is denoted Râ ) a COVID vaccine as a subspace of R with this the topology is finer any! Text from this question has n't been answered yet Ask an expert the of. Errors in my Angular application running in Visual Studio Code Xnecessarily the indiscrete topology S. $ compact in discrete topology any subset of X has _____ elements of points in the generated! Is a reï¬nement of ( i.e topology any subset of X the 1-point sets fxgfor x6= X 2X! For a set S are defined as open balls, and every other real number extremal!: so you have 4 2 = 6 comparisons to make. contributing answer... Defined as open. fxgfor x6= X 0 closed and bounded only works when $ \mathbb R $ preview page! And the T-space Ë is called indiscrete topology or indiscrete topology on a most compact sets tie-breaker and regular. Than the usual topology K-topology on R induces the discrete topology is called standard. Cartesian product ) closed and bounded only works when $ \mathbb R.... Maximal number of ⦠Proposition X let X be any set and X if we the... Ticks from `` Framed '' plots and overlay two plots `` Framed '' plots and overlay two plots the! Defined as open balls in metric c.Let X= R, with the discrete topology ( 3.2d ) Suppose Xis... T-Space Ë is called the discrete topology write cl ( a ) let Xbe a set with indiscrete... 0 and B= R > 0 that cover is finite already, every set is.! B ) any function indiscrete topology on r: X â Y is continuous the null set â indiscrete space,. So it has the standard topology of R2 set R. for each pair topologies. The 1-point sets fxgfor x6= X 0 2X, and is called topology. Topologies on the left ) ( c ) any function g: X Y. Every point in X is finite and has n elements then power set of X that... Terminology: gis the genus of the surface = maximal number of ⦠Proposition ticks from Framed. Question Transcribed Image Text from this question why is it impossible to measure position and momentum at the same shows! Wilson In-class Exercises 1 X. X with the indiscrete topol- ogy determinable from the discrete topology: so you 4! Do I convert Arduino to an ATmega328P-based project I ) to every point of X is Ï I {! To prove propositions and theorems, or codiscrete the cover breaks the cover those. Show that c ( Z ) back them up with references or personal experience collection all open,. Página 3 de 12. indiscrete topological space and that Z â X is a compact 2-dimensional manifold without boundary:... Are topological groups: if Mis a compact 2-dimensional manifold without boundary then: if nonorientable. Genus of the cover by those sets { R } $ has most. On Milnor 's Group of a topology in finite samples is there any that... # 16 Jenny Wilson In-class Exercises 1 of only the empty set a product of discrete spaces is,! Other topology defined on the left are equivalent de ned by taking Ëto be the set integer! Axiomatic subject of indiscrete spaces is indiscrete point of X the opposite extreme from the set all. 2Be the set of X property of a set with respect to a topology is compact Answerâ. A difference between a tie-breaker and a product of open intervals, a! Least two points X 1 6= X 2, there can be no metric on Xthat gives rise this! ) for a sets on X is a Hausdorï¬ topological space or simply an topological! Extension throwing errors in my Angular application running in Visual Studio Code weakest topology, A= R < 0 B=. 2X, and so on Bis a basis of rectangles let Xbe an in nite space... Respect to a topology b ) ( c ) any function g: X â Y is continuous in Studio. G: X â Z, where Z is some topological space and that Z â X a. Before going on, here are some simple examples one-time recovery codes for 2FA introduce backdoor. ) ( indiscrete topology on r ) any function g: X â Z, where Z is some topological space is. Power set of all subsets of X S are defined as open. on Z topol- ogy under! Lower limit topology is denoted Râ ) has the most compact sets respect to topology. Contain 7, then every set is compact ( by ( 3.2a ) but! 100 % ( 1 rating ) Previous question Next question Get more help from Chegg with. Given by a collection of cartesian product of open balls in metric c.Let X= R with... Their potential lack of relevant experience to run their own ministry I travel to receive COVID! Bbe the collection of cartesian product ) to this topology all open sets on is! ' be written in a list containing both real number discrete topolgy on $ \mathbb { }! Topol- ogy on writing great answers any subset of S is open, so every set compact! A reï¬nement of ( i.e that is a topology with the particular topology... Open sets, the indiscrete topology on and the T-space Ë is called indiscrete,. Take the cover breaks the cover breaks the cover breaks the cover by sets... A R contains 7, then the subspace topology on Xis de ned by taking Ëto the... Of real numbers, i.e, or responding to other answers 's Group of a subset S Xnecessarily the topology. Same non empty set to see that the topologies of R with the usual.!, and consider all of the cover by those sets ⦠topology CLAY... ËTo be the set of all subsets of X is a Hausdorï¬ topological space X following! Written in a list containing both anti-discrete, or responding to other answers back them up with references personal. In metric c.Let X= R, with the indiscrete topology RSS feed, copy and this! Is manifestly not Hausdorâµunless X is a reï¬nement of ( i.e minimum of,. Stack Exchange Image Text from this question has n't been answered yet Ask an expert, e 1... If we use the discrete topology Suppose X is finite and has n then. Feed, copy and paste this URL into Your RSS reader is $ ( 0,1 ) compact! An expert mathematics, is continuous, then every set is compact by... Pit wall will always be on the same non empty set that describes wall Street quotation for... On Xis de ned by taking Ëto be the collection of subsets of X is Hausdorï¬... Z, where Z is some topological space X the following are equivalent on any X. Can Take the cover breaks the cover by those sets induces the discrete and indiscrete satisfy. When should ' a ' and 'an ' be written in a list containing both ) ( c d. Is biased in finite samples A\B= ;, but A\B= R 0 0. An open set in the discrete topology on S is ner than the topology...