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I am new in studying Topology. So far, I am a bit confused in the purpose of using neighborhoods and not use just open sets (neighbourhoods).

A neighbourhood of a point $\textbf{x}$ is a set $N$ that contains an open set $U$ such that $\textbf{x} \in U$. In the proofs I have encountered so far, it was enough just to consider an open set containing a point $\textbf{x}$.

So, I wonder if neighbourhoods must contain an open set containing $\textbf{x}$, then why we need tu use notion of neighbourhoods in the first place and not use just open sets?

Thank you in advance!

Cezar98
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3 Answers3

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It allows for easier formulations of some facts, it's a handy shortcut. $ $$ $$ $

E.g. every point in $\Bbb R$ has a compact neighbourhood (e.g.$ $ $[x-1,x+1]$ will do) and even has a local base of compact neighbourhoods. If you would avoid the term neighbourhood we would formulate it as: every point $x$ is contained in an open set $O$ with compact closure, or which is contained in a compact set etc.

As another example, $X$ is regular (a property you might learn about later) iff every point has a base of closed neighbourhoods. Nice and short formulation.

Also continuity allows such a shorter pointwise formulation: $f$ is continuous in $x$ iff $f^{-1}[N]$ is a neighbourhood of $x$ for every neighbourhood $N$ of $f(x)$.

The set of neighbourhoods is automatically a so-called filter in $\mathscr{P}(X)$ which is used in theory of convergence (filters and nets etc.).

So it's a useful notion IMO. You could also choose to always use open sets, but that might take more words or give more cumbersome formulations.

aleph2
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Henno Brandsma
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We don't have to use or do anything. In fact most things in mathematics can be expressed in multiple ways. The choice of this "way" mostly depends on how convenient it is for the one using it.

And neighbourhoods is not an exception. Neighbourhoods describe the topology locally, around a single point. This is useful if we work with local properties. For example local compactness. And local properties are important because many spaces have them but do not have their global counterparts. For example $\mathbb{R}^n$ is locally compact but not compact. Any CW complex is locally contractible but not necessarily contractible. Any manifold is locally path connected but not necessarily path connected. And so on.

freakish
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Precisely as I was reading through this exchange, pushed by the unsatisfactory answer that my professor gave to a question identical to this one, it happened that he (during the lesson) gave a definition that's in line with Henno's formulations (where the concept of neighborhood is much welcomed).

A point $x$ belongs to the interior of $S$ if and only if $S$ is a neighbourhood of $x$.

At first I too thought of the concept of neighborhood as too cumbersome (who cares about those extra points?) but this most simple example changed my mind.

aleph2
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