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In the book "Topology" by James Munkres we define topology on a set $X$ like the following(page 76):

$\mathscr T$ is a topology on $X$ if:

  • $\emptyset,X\in\mathscr T$
  • $T\subset\mathscr T\implies\bigcup T\in\mathscr T$
  • $T\subset\mathscr T\land T\mbox{ is finite}\implies \bigcap T\in\mathscr T$

Why does the third property have the fact that $T$ is finite? What does it gives us?

ℋolo
  • 10,006
  • Because an infinite collection allows you to build a closed set from open sets, by intersection (for instance a single point from shrinking neighborhoods). –  Jun 11 '18 at 15:10
  • General definitions raise from concrete cases. If you take for example the standard topology on $\mathbb{R}$ then you will see that it does not preserver infinite intersections. The infinite version of (3) just isnt useful. – freakish Jun 11 '18 at 15:36

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