Product, Box, and Uniform Topologies 18 11. Typically, a discrete set is either finite or countably infinite. 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. De ne T indiscrete:= f;;Xg. A Theorem of Volterra Vito 15 9. Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. I think not, but the proof escapes me. If anything is to be continuous, it's the real number line. I mean--sure, the topology would have uncountably many subsets of the reals, but conceptually a discrete topology on the reals is possible, no? That is, T discrete is the collection of all subsets of X. Then consider it as a topological space R* with the usual topology. In: A First Course in Discrete Dynamical Systems. 52 3. discrete:= P(X). What makes this thing a continuum? The real number line [math]\mathbf R[/math] is the archetype of a continuum. $\endgroup$ – … 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. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. In nitude of Prime Numbers 6 5. Quotient Topology … Perhaps the most important infinite discrete group is the additive group ℤ of the integers (the infinite cyclic group). We say that two sets are disjoint Let Xbe any nonempty set. Consider the real numbers R first as just a set with no structure. 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 … $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. Product Topology 6 6. Another example of an infinite discrete set is the set . For example, the set of integers is discrete on the real line. The question is: is there a function f from R to R* whose initial topology on R is discrete? Topology of the Real Numbers In this chapter, we de ne some topological properties of the real numbers R and its subsets. TOPOLOGY AND THE REAL NUMBER LINE Intersections of sets are indicated by “∩.” A∩ B is the set of elements which belong to both sets A and B. Example 3.5. Therefore, the closure of $(a,b)$ is … Homeomorphisms 16 10. 5.1. Universitext. Compact Spaces 21 12. Then T discrete is called the discrete topology on X. 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