2

Rudin gives the definition of a base as follows in Exercise 2.23

A collection $\{V_\alpha\}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: for every $x\in X$ and every open set $G\subset X$ such that $x\in G$, we have that $x\in V_{\alpha}\subset G$ for some $\alpha$. In other words, every open set in $X$ is the union of a sub collection of $V_{\alpha}$

Munkres Topology gives the definition for a basis of a topological space as follows:

If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ (called basis elements) such that

  1. For each $x\in X$, there is at least one basis element $B$ containing $x$
  2. If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3\subset B_1 \cap B_2$

Are these two definitions (a base and a basis) equivalent? If so, how can I see that the Munkres's version is equivalent to the Rudin version?

random_0620
  • 2,241
  • 4
    Munkres definition is about the basis for a topology, meaning, a collection of subsets of $X$ that should work as a basis for some topology on $X$, in the sense of Rudin's definition. – azif00 Jun 06 '21 at 04:19
  • See my answer to Confusion Regarding Munkres's Definition of Basis for a Topology. – Dave L. Renfro Jun 06 '21 at 07:14
  • 1
    Note theat the commonly accepted term is "base" (as Rudin does), a "basis" is used in algebra mostly. Most papers I've ever read use base. – Henno Brandsma Jun 06 '21 at 08:45
  • @Henno Brandsma: Although I put a note about this near the beginning of the answer I cited, I think I should probably change the term to "base", and change the note to say that I'm using "base" instead of the other OP's term "basis". However, I don't feel up to doing this now, but hopefully I'll see this comment over the next couple of days and one of those times take care of things. – Dave L. Renfro Jun 06 '21 at 15:22

0 Answers0