Typically, a discrete set is either finite or countably infinite. In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology.For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. For example, the set of integers is discrete on the real line. The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. The question is: is there a function f from R to R* whose initial topology on R is discrete? Example 3.5. The real number field â, with its usual topology and the operation of addition, forms a second-countable connected locally compact group called the additive group of the reals. Product, Box, and Uniform Topologies 18 11. A Theorem of Volterra Vito 15 9. 5.1. If $\tau$ is the discrete topology on the real numbers, find the closure of $(a,b)$ Here is the solution from the back of my book: Since the discrete topology contains all subsets of $\Bbb{R}$, every subset of $\Bbb{R}$ is both open and closed. Product Topology 6 6. Therefore, the closure of $(a,b)$ is â¦ Another example of an infinite discrete set is the set . The points of are then said to be isolated (Krantz 1999, p. 63). We say that two sets are disjoint I mean--sure, the topology would have uncountably many subsets of the reals, but conceptually a discrete topology on the reals is possible, no? $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? Compact Spaces 21 12. If anything is to be continuous, it's the real number line. $\endgroup$ â â¦ Then T discrete is called the discrete topology on X. 52 3. Then consider it as a topological space R* with the usual topology. The real number line $\mathbf R$ is the archetype of a continuum. Let Xbe any nonempty set. TOPOLOGY AND THE REAL NUMBER LINE Intersections of sets are indicated by ââ©.â Aâ© B is the set of elements which belong to both sets A and B. I think not, but the proof escapes me. In nitude of Prime Numbers 6 5. Subspace Topology 7 7. De ne T indiscrete:= f;;Xg. Open sets Open sets are among the most important subsets of R. A collection of open sets is called a topology, and any property (such as â¦ Quotient Topology â¦ Topology of the Real Numbers In this chapter, we de ne some topological properties of the real numbers R and its subsets. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. 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