Every T 1 space is T 0. Example. Meaning of indiscrete with illustrations and photos. Example of a topological space with a topology different from the discrete and indiscrete one with identical clopen sets. for each x,y ∈ X such that x 6= y there is an open set U ⊂ X so that x ∈ U but y /∈ U. T 1 is obviously a topological property and is product preserving. coarsest possible topology on Xis the indiscrete topology on X, which has as few open sets as possible: only ;and Xare open (think of a monitor which can only display a solid eld of black or white). valid topology, called the indiscrete topology. But then $U_j \not \subseteq X$ for all $j \in \{ 1, 2, ..., n \}$ which contradicts the fact that $U_1, U_2, ..., U_n$ are a collection of subsets of $\mathcal P(X)$. (1) The usual topology on the interval I:= [0,1] ⊂Ris the subspace topology. $\displaystyle{\bigcup_{i \in I} U_i \in \tau}$, $\displaystyle{\bigcap_{i=1}^{n} U_i \in \tau}$, $\mathcal P(X) = \{ Y : Y \subseteq X \}$, $\displaystyle{\bigcup_{i \in I} U_i \not \in \mathcal P(X)}$, $\displaystyle{\bigcup_{i \in I} U_i \not \subseteq X}$, $\displaystyle{x \in \bigcup_{i \in I} U_i}$, $\mathcal P(X) = \{ U : U \subseteq X \}$, $\displaystyle{\bigcup_{i \in I} U_i \in \mathcal P(X)}$, $\displaystyle{\bigcap_{i=1}^{n} U_i \not \in \mathcal P(X)}$, $\displaystyle{\bigcap_{i=1}^{n} U_i \not \subseteq X}$, $\displaystyle{x \in \bigcap_{i=1}^{n} U_i}$, $\displaystyle{\bigcap_{i=1}^{n} U_i \in \mathcal P(X)}$, $\emptyset \cup \emptyset = \emptyset \in \{ \emptyset, X \}$, $\emptyset \cup X = X \in \{ \emptyset, X \}$, $\emptyset \cap \emptyset = \emptyset \in \{ \emptyset, X \}$, $\emptyset \cap X = \emptyset \in \{ \emptyset, X \}$, Creative Commons Attribution-ShareAlike 3.0 License. The indsicrete topology is defined as follows: Let X be a non-empty set and let T be the collection of the empty set ( ϕ) and the set X. i.e T = { ϕ, X }, if T is a topology on X, then such a topology is called an indiscrete topology and the pair ( X, T) is called an indiscrete topological space. As per the corollary, every topology on X must contain \emptyset and X, and so will feature the trivial topology as a subcollection. 1 2 ALEX KURONYA The first topology in the list is a common topology and is usually called the indiscrete topology; it contains the empty set and the whole space X. Wolfram Web Resources. Page 1. Prove that Xis compact. The following examples introduce some additional common topologies: Example 1.4.5. Definition of indiscrete in the Fine Dictionary. 4. Then $U_j \not \subseteq X$, which contradicts the fact that $\{ U_i \}_{i \in I}$ is an arbitrary collection of subsets from $\mathcal P(X) = \{ U : U \subseteq X \}$. 7. I hope you are all understand the concept of discrete topology and indiscrete topology. Also, it is understood that ∅ is in all the topologies.? Pronunciation of indiscrete and its etymology. This preview shows page 1 - 2 out of 2 pages. Click here to edit contents of this page. Note: The topology which is both discrete and indiscrete such topology which has one element in set X. i.e. Then Xis compact. De nition 1.1.1 A collection ˝of subsets of Xis said to de ne a topology on Xif it satis es the following three conditions. For a trivial example, let X be an infinite set with the indiscrete topology; consider the singletons of X. Topology induced by a map. R under addition, and R or C under multiplication are topological groups. In general in any space with 2 or more poinys that has the indiscrete topology (thus only nothing and everything are open sets), no singelton is closed. Most people chose this as the best definition of discrete-topology: (mathematics) A topology... See the dictionary meaning, pronunciation, and sentence examples. (3) The sphere Snis the subspace Sn⊂Rn+1 of points of norm one. De nition 13. Proving Willard Theorem 3.11 .. [25 points] (i) Give an example of a nonmetrizable space (in other words a topological space (X, U) which is not the underlying topological space for some metric space (X, d)). 21 November 2019 Math 490: Worksheet #16 Jenny Wilson In-class Exercises 1. Example sentences containing indiscrete If d is a metric on T , the collection of all d-open sets is a topology on T . Reviews. Example sentences with "indiscrete topology", translation memory. Today i will be giving a tutorial on the discrete and indiscrete topology, this tutorial is for MAT404(General Topology), Now in my last discussion on topology, i talked about the topology in general and also gave some examples, in case you missed the tutorial click here to be redirect back. The indiscrete topology on a set Xis de ned as the topology which consists of the subsets ? 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. School Western University; Course Title MATH 2122; Uploaded By jguo246. • Every two point co-finite topological space is a $${T_1}$$ space. Let (X;T X) be a topological space. So $x \in U_j$ for all $j \in \{1, 2, ..., n \}$. Ask Question Asked today. for some n2N. This is a valid topology, called the indiscrete topology. 1.3. Give an example of a set X and two topologies T1 and T2 for X such that TUT2 is not a topology for X. (For any set X, the collection of all subsets of X is also a topology for X, called the "discrete" topology. because it contains only ∅ and?.? Example 2. • An indiscrete topological space with at least two points is not a $${T_1}$$ space. Example 2. Similarly, if Xdisc is the set X equipped with the discrete topology, then the identity map 1 X: Xdisc!X 1 is continuous. MA222 – 2008/2009 – page 2.1 Example 1.3. Unless otherwise stated, the content of this page is licensed under. For the third condition, the only possible intersections are $\emptyset \cap \emptyset = \emptyset \in \{ \emptyset, X \}$, $\emptyset \cap X = \emptyset \in \{ \emptyset, X \}$, and $X \cap X = X \in \{ \emptyset, X \}$. To wrap up today, let’s talk about one more example of a topology. If Xhas at least two points x 1 6= x 2, there can be no metric on Xthat gives rise to this topology. X = {a}, $$\tau = $${$$\phi $$, X}. Example 1.4. Metric spaces have a metric which is positive-de nite, symmetric and satis es the triangle inequality. add example. The metric is called the discrete metric and the topology is called the discrete topology. The properties verified earlier show that is a topology. Any indiscrete space is perfectly normal (disjoint closed sets can be separated by a continuous real-valued function) vacuously since there don't exist disjoint closed sets. For the second condition, the only possible unions are $\emptyset \cup \emptyset = \emptyset \in \{ \emptyset, X \}$, $\emptyset \cup X = X \in \{ \emptyset, X \}$, and $X \cup X = X \in \{ \emptyset, X \}$. The indiscrete nucleus does not have a nuclear membrane and is therefore not separate from the cytoplasm. Indiscrete definition: not divisible or divided into parts | Meaning, pronunciation, translations and examples In topology: Topological space …set X is called the discrete topology on X, and the collection consisting only of the empty set and X itself forms the indiscrete, or trivial, topology on X.A given topological space gives rise to other related topological spaces. Here, the notation "$\mathcal P(X) = \{ Y : Y \subseteq X \}$" represents the power set of $X$ or rather, the set of all subsets of $X$. Unless someone's been indiscrete. Let X be an infinite set and let $\mathcal T $ be a topology on X. For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable set. In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the fewest possible open sets (just the empty set and the space itself).Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space is continuous, etc. Meaning of indiscrete with illustrations and photos. This is not obvious at all, but we will prove it shortly. Now let Ube any open cover of !+ 1, and let U Let f : X!Y be a map of sets. The is a topology called the discrete topology. [0;1] with its usual topology is compact. 4. ⇐ Definition of Topology ⇒ Indiscrete and Discrete Topology ⇒ One Comment. Some sample topologies: (1)Discrete topology: T= 2X. The "indiscrete" topology for any given set is just {φ, X} which you can easily see satisfies the 4 conditions above. For example, a … Good to hear from you. Related words - indiscrete synonyms, antonyms, hypernyms and hyponyms. Only ∅ and T are open. 4 is called the discrete topology on?, as it contains every subset of?. en If S = (0,1) is the open unit interval, a subset of the real numbers, then 0 is a condensation point of S. If S is an uncountable subset of a set X endowed with the indiscrete topology, then any point p of X is a condensation point of X as the only open neighborhood of p is X itself. (xii)If a sequence of points (a n) n2N in a topological space Xconverges to a point a 1, then a 1is a limit point of the set fa njn2Ng. But on the other hand, the only T0 indiscrete spaces are the empty set and the singleton.. Metrizability. this is called the codiscrete topology on S S (also indiscrete topology or trivial topology or chaotic topology), it is the coarsest topology on S S; Codisc (S) Codisc(S) is called a codiscrete space. Suppose that $\displaystyle{\bigcup_{i \in I} U_i \not \in \mathcal P(X)}$. See pages that link to and include this page. X with the indiscrete topology is called an indiscrete topological space or simply an indiscrete space. Let Xbe a topological space with the indiscrete topology. The induced topology is the indiscrete topology. which equips a given set with the indiscrete topology. Append content without editing the whole page source. 0 but indiscrete spaces of more than one point are not T 0. (Oscar Wilde, An Ideal Husband ) (b) Topology aims to formalize some continuous, _____ features of space. In this video you will learn about topological space types , Discrete and indiscrete topologies , trivial topology , strongest and smallest topology....with best Explaination....examples … (2)Indiscrete topology: T= f?;Xg. 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. For an axiomatization of this situation see codiscrete object. Definition of indiscrete in the Fine Dictionary. Example 1. Now consider any arbitrary collection of subsets $\{ U_i \}_{i \in I}$ from $\mathcal P(X)$ for some index set $I$. Some "extremal" examples Take any set X and let = {, X}. 1 2 ALEX KURONYA The first topology in the list is a common topology and is usually called the indiscrete topology; it contains the empty set and the whole space X. It is called the indiscrete topology or trivial topology. R and C are topological elds. Separation properties. Topology has several di erent branches | general topology (also known as point-set topology), algebraic topology, di erential topology and topological algebra | the rst, general topology, being the door to the study of the others. Let (X;T X) be a topological space. For the first condition, we clearly see that $\emptyset \in \{ \emptyset, X \}$ and $X \in \{ \emptyset, X \}$. Practice (a) "Questions are never _____; answers sometimes are." Practice (a) "Questions are never _____; answers sometimes are." Wolfram|Alpha » Explore anything with the first computational knowledge engine. Then the sequence converges to both xand to y. Example1.23. R is disconnected with the subspace topology. Definition: If $X$ is any set, then the Indiscrete Topology on $X$ is the collection of subsets $\tau = \{ \emptyset, X \}$. 2) prove that if $\mathcal T $ contains every infinite subset of X, then it is the indiscrete topology. Let {I α | α ∈ A} be an infinite collection of segments I α = [0, 1]. Example (Indiscrete topologies). Watch headings for an "edit" link when available. WikiMatrix. Pronunciation of indiscrete and its etymology. With such a restrictive topology, such spaces must be examples/counterexamples for … (ii) State which of the following statements is/are true and which is/are false.Reasons are not needed for correct answers, but for incorrect answers they may yield partial credit. View/set parent page (used for creating breadcrumbs and structured layout). Indiscrete Topology 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. JavaScript is disabled. También, cualquier conjunto puede ser dotado de la topología trivial (también llamada topología indiscreta ), en la que sólo el conjunto vacío y … (ii)The other extreme is to take (say when Xhas at least 2 elements) T = f;;Xg. Say that $x \in U_j$ for some $j \in I$. (Oscar Wilde, An Ideal Husband ) (b) Topology aims to formalize some continuous, _____ features of space. Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top. Topology induced by a map. Example in topology: quotient maps and arcwise connected. Mathematica » The #1 tool for creating Demonstrations and anything technical. However: The usual topology is the smallest topology containing the upper and lower topology. Click here to toggle editing of individual sections of the page (if possible). Indiscrete topology or Trivial topology - Only the empty set and its complement are open. Indiscrete Topology. The discrete topology is just 풫(?) Let Xbe a set. The closed sets are the complements of those, which are {1, 2} and {}. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every singleton set is discrete as well as indiscrete topology on that set. View wiki source for this page without editing. … Then is a topology called the trivial topology or indiscrete topology. Then Xis not compact. (3)The induced topology on a metric space. Since $\emptyset \subseteq X$ and $X \subseteq X$, we clearly have that $\emptyset, X \subseteq \mathcal P(X)$, so the first condition holds. topologies for 3. Then $\displaystyle{\bigcap_{i=1}^{n} U_i \not \subseteq X}$, so there exists an $\displaystyle{x \in \bigcap_{i=1}^{n} U_i}$ such that $x \not \in X$. Let's verify that $(X, \tau) = (X, \mathcal P(X))$ is indeed a topological space. compact (with respect to the subspace topology) then is Z closed? Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social … There’s a forgetful functor [math]U : \text{Top} \to \text{Set}[/math] sending a topological space to its underlying set. I aim in this book to provide a thorough grounding in general topology… If you want to discuss contents of this page - this is the easiest way to do it. For example, the collection of all subsets of a set X is called the discrete topology on X, and the collection consisting only of the empty set and X itself forms the indiscrete, or trivial, topology on X. Several other "Counterexamples in ..." books and papers have followed, with similar motivations. 2. As you can see, neither of the one-point sets {1} or {2} is open or closed. Related words - indiscrete synonyms, antonyms, hypernyms and hyponyms. In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. 2011. independent; induce; Look at other dictionaries: topology — topologic /top euh loj ik/, topological, adj. Page 1. T= fU X: 8x2U9 s:t:O (x) Ug. For an example in a more familiar setting, let X be the real line with its usual topology; then each point of X is in at most one of the open intervals [ 1 n + 1 , 1 n ] (for integers n > 0), but any neighborhood of 0 contains infinitely many of those intervals. This functor has both a left and a right adjoint, which is slightly unusual. Set Theory, Logic, Probability, Statistics, Effective planning ahead protects fish and fisheries, Polarization increases with economic decline, becoming cripplingly contagious. Then Xis compact. Then $\displaystyle{\bigcup_{i \in I} U_i \not \subseteq X}$ and so there exists an element $\displaystyle{x \in \bigcup_{i \in I} U_i}$ such that $x \not \in X$. (the power set of? Check out how this page has evolved in the past. and Xonly. Example (Topology induced by a metric). Some sample topologies: (1)Discrete topology: T= 2X. Remark Therefore $\displaystyle{\bigcup_{i \in I} U_i \in \mathcal P(X)}$. Example 1.4. Properties. We check that the topology B generated by B is the VIP topology on X:Let U be a subset of Xcontaining p:If x2U then choose B= fpgif x= p, In the. Example 1.5. Thus the 1st countable normal space R 5 in Example II.1 is not metrizable, because it is not fully normal. Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set). Then τ is a topology on X. X with the topology τ is a topological space. For an example in a more familiar setting, let X be the real line with its usual topology; then each point of X is in at most one of the open intervals [ 1 n + 1 , 1 n ] (for integers n > 0), but any neighborhood of 0 contains infinitely many of those intervals. Find out what you can do. Example sentences with "indiscrete topology", translation memory. Hope you're managing OK in the current difficult times. Counter-example topologies [ edit ] The following topologies are a known source of counterexamples for point-set topology . The indiscrete topology on a set Xis de ned as the topology which consists of the subsets ? (Do not use the indiscrete topology.) topologically, adv. Pages 2. Let τ be the collection all open sets on X. on R:The topology generated by it is known as lower limit topology on R. Example 4.3 : Note that B := fpg S ffp;qg: q2X;q6= pgis a basis. (a) Let Xbe a set with the co nite topology. It is the topology associated with the discrete metric. indiscrete) is compact. 1.3. With such a restrictive topology, such spaces must be examples/counterexamples for many other topological properties. Notify administrators if there is objectionable content in this page. 8. No translation memories found. Every metric space (X;d) has a topology which is induced by its metric. No! For example, consider X = fx;ygwith the indiscrete topology. 21 November 2019 Math 490: Worksheet #16 Jenny Wilson In-class Exercises 1. Then for every … Metric spaces have a metric which is positive-de nite, symmetric and satis es the triangle inequality. All subsets of T are open. English-Finnish mathematical dictionary. Let Xbe an in nite topological space with the discrete topology. Let S S be a set and let (X, τ) (X,\tau) be a topological space. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In this video you will learn about topological space types , Discrete and indiscrete topologies , trivial topology , strongest and smallest topology....with best Explaination....examples … Examples of topological spaces The discrete topology on a set Xis de ned as the topology which consists of all possible subsets of X. ). Definition 2.2 A space X is a T 1 space or Frechet space iff it satisfies the T 1 axiom, i.e. For example take X to be a set with two elements α and β, so X = {α,β}. indiscrete topology. In topology: Topological space …set X is called the discrete topology on X, and the collection consisting only of the empty set and X itself forms the indiscrete, or trivial, topology on X. Example sentences containing indiscrete Then GL(n;R) is a topological group, and … General Wikidot.com documentation and help section. I don't think I agree with (e) that one-point sets are closed. Then ρ is obviously compatible with the discrete topology of X. (3)The induced topology on a metric space. Let X be the set of points in the plane shown in Fig. Something does not work as expected? Therefore $\displaystyle{\bigcap_{i=1}^{n} U_i \in \mathcal P(X)}$. On the other hand, a metrizable space must have all topological properties possessed by a metric space. Any group given the discrete topology, or the indiscrete topology, is a topological group. Example the indiscrete topology on x is τ i x every. SEE: Trivial Topology. 5. Wikidot.com Terms of Service - what you can, what you should not etc. We check that the topology B generated by B is the VIP topology on X:Let U be a subset of Xcontaining p:If x2U then choose B= fpgif x= p, and B= fp;xgotherwise. Let $X$ be a nonempty set and let $\tau = \{ \emptyset, X \}$. For a better experience, please enable JavaScript in your browser before proceeding. To see this, rst recall that we have already seen that any nontrivial basic open set containing the top point !must be of the form (n;1) = (n;!] This is known as the trivial or indiscrete topology, and it is somewhat uninteresting, as its name suggests, but it is important as an instance of how simple a topology may be. ; An example of this is if " X " is a regular space and " Y " is an infinite set in the indiscrete topology. A given topological space gives rise to other related topological spaces. Since all three conditions for $\tau = \{ \emptyset, X \}$ hold, we have that $(X, \{ \emptyset, X \})$ is a topological space. Since all three conditions for $\tau = \mathcal P(X)$ hold, we have that $(X, \mathcal P(X))$ is a topological space. Important fundamental notions soon to come are for example open and closed sets, continuity, homeomorphism. Every metric space (X;d) has a topology … I make a video on the concept of discrete and indiscrete topology. Prove that T is the discrete topology for X iff every subset consisting of one point is open. Then Xis not compact. Important fundamental notions soon to come are for example open and closed sets, continuity, homeomorphism. and Xonly. If Xhas at least two points x 1 6= x 2, there can be no metric on Xthat gives rise to this topology. 3. Then. Let Xbe a topological space with the indiscrete topology. Interesting topologies are balanced between these two extremes. Then Z = {α} is compact (by (3.2a)) but it is not closed. Give an example of a topology on an infinite set which has only a finite number of elements. This particular counterexample shows that second-countability does not follow from first-countability. 1 is called the trivial topology (or indiscrete topology) on? View and manage file attachments for this page. 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. In the indiscrete topology the only open sets are φ and X itself. on R:The topology generated by it is known as lower limit topology on R. Example 4.3 : Note that B := fpg S ffp;qg: q2X;q6= pgis a basis. Let Xbe an in nite topological space with the discrete topology. Geometry - Topology; What is the difference? 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. Regard X as a topological space with the indiscrete topology. Interior and Closure in a Topological Space ... ... remark by Willard. False. I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 1: Set Theory and Metric Spaces and am currently focused on Section 2: Metric Spaces ... ... Can you remind us of the meaning of "Pseudometrizable" and "Pseudo metric"? The Discrete Topology For example, consider the constant sequence (0) n2N in R. Then the sequence converges to Lastly, consider any finite collection of subsets $U_1, U_2, ..., U_n$ of $\mathcal P(X)$. Indiscrete definition: not divisible or divided into parts | Meaning, pronunciation, translations and examples Example 3. Recall from the Topological Spaces page that a set $X$ and a collection $\tau$ of subsets of $X$ together denoted $(X, \tau)$ is called a topological space if: We will now look at two rather trivial topologies known as the discrete topologies and the indiscrete topologies. Example: The indiscrete topology on X is τ I = {∅, X}. 6. (2) The set of rational numbers Q ⊂Rcan be equipped with the subspace topology (show that this is not homeomorphic to the discrete topology). The indiscrete nucleus does not have a nuclear membrane and is therefore not separate from the cytoplasm. I agree with this. • The discrete topological space with at least two points is a $${T_1}$$ space. 4. Under the trivial topology, the open sets are {} and {1, 2}. False. Interpretation Translation indiscrete topology. A given topological space gives rise to other related topological spaces. minitopologia. T= fU X: 8x2U9 s:t:O (x) Ug. Example 1.3. Suppose that $\displaystyle{\bigcap_{i=1}^{n} U_i \not \in \mathcal P(X)}$. Example 1.5. 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"). The subspace topology provides many more examples of topological spaces. Change the name (also URL address, possibly the category) of the page. i think this is untrue, Let X be any set and let be the set of all subsets of X. !+ 1 is compact. Math. Consider where X = {1, 2}. For example, a subset A of a topological space X … ; The greatest element in this fiber is the discrete topology on " X " while the least element is the indiscrete topology. It consists of all subsets of Xwhich are open in X. Indiscrete Topology: Eric Weisstein's World of Mathematics [home, info] indiscrete topology: PlanetMath Encyclopedia [home, info] Words similar to indiscrete topology Usage examples for indiscrete topology Words that often appear near indiscrete topology Rhymes of indiscrete topology Invented words related to indiscrete topology: Search for indiscrete topology on Google or Wikipedia. Hello Peter. Prove that for any nonempty set $X$ that if $\tau$ is the indiscrete topology then $(X, \tau)$ is not a Hausdorff space. Example (Discrete topologies). topology 1.1 Some de nitions and examples Let Xbe a set. topologist, n. /teuh pol euh jee/, n., pl. For a trivial example, let X be an infinite set with the indiscrete topology; consider the singletons of X. Every sequence converges in (X, τ I) to every point of X. (Limits of sequences are not unique.) One again, let's verify that $(X, \tau) = (X, \{ \emptyset, X \})$ is indeed a topological space. A space $X$ is indiscrete provided its topology is $\{\emptyset,X\}$. También, cualquier conjunto puede ser dotado de la topología trivial (también llamada topología indiscreta), en la que sólo el conjunto vacío y el espacio en su totalidad son abiertos. (2)Indiscrete topology: T= f?;Xg. (See Example III.3.) 6. (a) Let Xbe a set with the co nite topology. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. add example. 7. Let Rbe a topological ring. Examples of topological spaces The discrete topology on a set Xis de ned as the topology which consists of all possible subsets of X. • Every two point co-countable topological space is a $${T_1}$$ space. Let’s look at points in the plane: [math](2,4)[/math], [math](\sqrt{2},5)[/math], [math](\pi,\pi^2)[/math] and so on.
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