3

From wikipedia

If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$, $$p \in U \subseteq V.$$

What is the added advantage of defining the neighborhood to be $V$, which need not be open, as opposed to defining the neighborhood to be the open set $U$ from the start?

Henno Brandsma
  • 250,824
yoyostein
  • 20,428
  • 4
    Some geometer wants to talk about something like "compact neighborhood centered at x", and then we have to define some non-open neighborhood(otherwise one may suffer from cumbersome notation like "compact subset $K$ whose interior $K^\circ$ includes $x$" whenever required.) – cjackal Apr 17 '16 at 15:52
  • There are treatments in topology that start with the basic notion of a neighborhood and define open sets (I like this approach). The wikipedia article mentions this further down in the article. – shoda Apr 17 '16 at 16:19

1 Answers1

3

The short answer: it's convenient.

E.g. one has the definition of local compactness: $X$ is locally compact if every point has a base of compact neighbourhoods. Or another (non-equivalent in general!) definition of local compactness: every point has a compact neighbourhood.

Or (theorem:) a space is regular iff it every point has a base of closed neighbourhoods.

Such definitions or theorems are easier to state with a general notion of neighbourhood. Also, the set of all neighbourhoods is then a filter while the open neighbourhoods are "just" a filter base, etc. This will only appeal to set theory minded people (like myself), probably. We can still say "open neighbourhood of $x$" for an open set that contains $x$, so it's easy to speciaise to that case as well.

For many definitions (like convergence, continuity) it does not matter whether we state them in terms of open neighbourhoods or general neighbourhoods. S owe can be a bit more general using the general term as well.

Henno Brandsma
  • 250,824