Intuition under Neighbourhood concept in Topological Spaces

Question

I sense that I didn't grasp well the concept of “Neighbourhood” in topological spaces. I did read through definitions of neighborhood, open and closed spaces. Despite that, I miss the utility of such a concept. For example, in the definition of limit of a sequence in a topological space: a point $x$ of the topological space $(X, \tau)$ is the limit of the sequence $(x_n)$ if, for every neighborhood $U$ of $x$, there is an $N$ such that, for every $n\geq N$, $x_n$ is in $U$. I don't see why we choose to express the limit of $(x_n)$ in terms of neighborhoods of $x$, why not an open set on $x$?

I would be glad if someone clarify this for me please. Thanks!

A neighborhood of $x$ is an open set containing $x$. What do you mean by an open set "on" $x$? — Ethan Alwaise, Feb 24 '16 at 23:57
@Ethan: this definition is not universal, unfortunately. Some don't demand that a neighborhood is open; only that it contains an open subset, which contains the specified point. — Eric Stucky, Feb 25 '16 at 00:19

score 1 · Answer 1 · answered Feb 25 '16 at 00:05

Consider the metric space $(X,d)$. We say $x_n \to x$ is $\forall \epsilon>0, \exists N \in \mathbb{N}$ such that $d(x_n,x)< \epsilon$ for all $n >N$ i.e for all $\epsilon >0$ you can find an $N$ such that $x_n \in B(x, \epsilon)$.

In general $(X, \tau)$ be not be a metric space. However, the above definition is equivalent to say that there exists an index $N$ such that for all $n> N$ we have $x_n \in U_x$ where $U_x \subset \tau$. Here $U_x$ is the equivalent of the ball.

Just keep in mind that for a metric space, neighborhoods i.e open sets are epsilon balls.

score 1 · Accepted Answer · answered Feb 25 '16 at 13:07

You actually do not need all three concepts (open, closed, and neighborhood); you can any two of them using the third. But they have different flavors, and they are very useful to express particular notions without needing to retort your speech for the only sake of using just one basic concept.

For example, you understand that you may express convergence but using only open sets. But neighborhoods (defined as sets containing an open set around a point) have the convenient property of being closed upwards, meaning that if $V$ is a neighborhood of $x$ and $W\supseteq V$ then $W$ also is. For instance, every closed square is a neighborhood of its center. Hence, you immediately have that given any closed square $S$ centered at $x$ and a sequence $(x_n)_{n\in\mathbb{N}}$ converging to $x$, then its tail is included in $S$. This might be of use if the problem at hand is better phrased in terms of closed squares.

On the other hand, the axioms of a topological space can be presented in a very neat way when you use open sets. An example of property which is better expressed by open sets is separation: Two subsets are separated if they are contained in disjoint open sets. (Note that the concept of “neighborhood” is most often followed by ”of a point”, not for subsets.)

Intuition under Neighbourhood concept in Topological Spaces

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