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  • Definition of discrete topology - Mathematics Stack Exchange
    Q1) Does discrete topology has to have all sets as open? Q2) I was trying to understand, how this discrete topology is the finest strongest form of topology and all the rest of the topologies lie in between discrete and ϕ ϕ We can even define a topological space which includes open and closed sets, would not this be more finer than discrete
  • Why is a discrete topology called a discrete topology?
    The discrete topology is the finest topology—it cannot be subdivided further If you think of the elements of the set as indivisible "discrete" atoms, each one appears as a singleton set You can effectively "see" the individual points in the topology itself
  • The difference between standard topology and discrete topology (both on . . .
    Also note that in the discrete topology every singleton $\{x\} \subseteq \mathbb{R}$ is open in $\mathbb{R}$
  • Discrete Topology and Closed Sets - Physics Forums
    I am trying to learn some topology and was looking at a problem in the back of the book asking to show that a topological space with the property that all set are closed is a discrete space which, as understand it, means that all possible subsets are in the topology and since all subsets are closed for each set in the topology the compliment
  • Why are singletons open in a discrete topology?
    A topological space having the discrete topology is called a discrete space The discrete topology is the finest topology on a set Right before this example, the author provides a lemma called local criterion for openness, but I am unable to see why a singleton in this situation is open
  • Prove that a topology Ƭ on X is the discrete topology if and only if . . .
    Here's how the authors define a topology: Let X be a set A topology Ƭ on X is a collection of subsets of X, each called an open set, such that (i) ∅ and X are open sets; (ii) The intersection of finitely many open sets is an open set; (iii) The union of any collection of open sets is an open set Here's how they define a discrete topology:
  • Topology: Prove that this subspace topology is discrete
    The question is from Topology and Its Applications Chapter 1, by William F Basner The question states the following, Let $\mathbb{Z}$ be a topological space with subspace topology inherited from $\mathbb{Z} \subset \mathbb{R}$ Prove that $\mathbb{Z}$ has discrete topology Proof Since $\mathbb{Z} \subset \mathbb{R}$, we note that from the
  • The bases for the discrete topology - Mathematics Stack Exchange
    The collection $\mathcal{B} = \{ \{x\} : x \in X \}$ is a basis for the discrete topology on a set X If X is a finite set with n elements, then clearly $\mathcal{B}$ also has n elements Is there a basis with fewer than n elements that generates the discrete topology on X?
  • What is, exactly, a discrete group? - Mathematics Stack Exchange
    Any set can be given the discrete topology, and any group can be given the discrete topology to make it a discrete group $\endgroup$ – Qiaochu Yuan Commented Jun 17, 2011 at 17:26
  • Product topology is discrete - Mathematics Stack Exchange
    Then $\prod_{\alpha\in A}X_{\alpha}$(under product topology) will be a discrete space if and only if $\{\alpha \in A: |X_\alpha| > 1\}$ is finite Or as an escape clause: maybe Willard defines somewhere that a discrete space is by definition one that has at least 2 points, or some such trick





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