For example, the set of integers is discrete on the real line. 52 3. The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. Example 3.5. Product, Box, and Uniform Topologies 18 11. If anything is to be continuous, it's the real number line. A Theorem of Volterra Vito 15 9. Compact Spaces 21 12. 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 ⦠Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. Subspace Topology 7 7. 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. 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. What makes this thing a continuum? The points of are then said to be isolated (Krantz 1999, p. 63). Topology of the Real Numbers In this chapter, we de ne some topological properties of the real numbers R and its subsets. Continuous Functions 12 8.1. Then consider it as a topological space R* with the usual topology. That is, T discrete is the collection of all subsets of X. The question is: is there a function f from R to R* whose initial topology on R is discrete? Another example of an infinite discrete set is the set . Then T discrete is called the discrete topology on X. I think not, but the proof escapes me. Quotient Topology ⦠$\endgroup$ â ⦠$\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? In nitude of Prime Numbers 6 5. Typically, a discrete set is either finite or countably infinite. 5.1. A set is discrete in a larger topological space if every point has a neighborhood such that . 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. Homeomorphisms 16 10. Therefore, the closure of $(a,b)$ is ⦠Product Topology 6 6. The real number line [math]\mathbf R[/math] is the archetype of a continuum. Consider the real numbers R first as just a set with no structure. Universitext. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. 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