I've just started looking at Rudin (Real and complex analysis, the big one) in which he offers the following definition (1.2):(adsbygoogle = window.adsbygoogle || []).push({});

a)A collection [tex] \tau[/tex] of subsets of a set X is said to be atopology in Xif [tex] \tau [/tex] has the following three properties:

i) [tex]\emptyset \in \tau [/tex] and [tex]X \in \tau [/tex]

ii) If [tex]V_{i}\in\tau[/tex] for i=1,[tex] \cdots [/tex],n, then [tex]V_{1} \cap V_{2} \cap \cdots \cap V_{n} \in \tau[/tex]

iii) If {[tex]V_{a}[/tex]} is an arbitrary collection of members of [tex]\tau[/tex] (finite, countable or uncountable) then [tex]\bigcup_{\alpha}V_{\alpha}\in \tau[/tex].

b)if [tex] \tau[/tex] is a topology in X, then X is called a topological space, and the members of X are called the open sets in X.

(Apologies for the random superscripts, I'm not sure why LaTeX felt they were necessary).

By considering the set {X, [tex]\emptyset[/tex]} doesn't this make every set a topological space? The first condition is obviously satisfied, the union of the empty set and any other set X is X, and the intersection of X with the empty set is the empty set... right?

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# Trivial topology on an arbitrary set?

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