nLab convergence

This entry is about the notion of limit in analysis and topology. For the notion of the same name in category theory see at limit.

Idea

A limit of a sequence (or net) of points $(x_i)$ in a topological space (or other convergence space) $X$ is a point $x$ such that the sequence eventually gets arbitrarily close to $x$. We can also speak of a limit of a filter on $X$.

The notion is of particular and historical importance in analysis, where it serves to define for instance the notion of derivative.

Definitions

The precise definition depends on what sort of space $X$ is.

Definition

If $X$ a topological space and $I$ the set of natural numbers (or more generally any directed set) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a sequence (or a net) of points in $X$, one says that a point $x \in X$ is a limit of $\nu$ or that $\nu$ converges to $x$ if for each neighbourhood $U$ in $X$ of $x$ there exists an $n \in I$ such that $x_i \in U$ for each $i \geq n$.

An important special case (the original) is:

Definition

If $X$ the real line and $I$ the set of natural numbers (or more generally any directed set) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a sequence (or a net) of real numbers, one says that a point $x \in X$ is a limit of $\nu$ or that $\nu$ converges to $x$ if for each positive number $\epsilon$ there exists an $n \in I$ such that ${|x_i - x|} \lt \epsilon$ for each $i \geq n$.

An important generalization (possibly the most general) is:

Definition

If $X$ a convergence space and $I$ the set of natural numbers (or more generally any directed set) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a sequence (or a net) of points in $X$, one says that a point $x \in X$ is a limit of $\nu$ or that $\nu$ converges to $x$ if the eventuality filter of $\nu$ converges to $x$ (which is a primitive concept in convergence spaces).

Other types of space for which we might put in definitions (or which might have definitions on their own pages) are (extended) (quasi)-(pseudo)-metric spaces, premetric spaces, (quasi)-uniform spaces, pretopological spaces, and (quasi)-uniform convergence spaces.

Notation

When one of the conditions above holds, we may write any of the following, where ‘$\to$’ is read as ‘converges to’:

• $\nu \to x$,
• $x_i \to_i x$,
• $x_i \to x$ (suppressing the index $i$ by abuse of notation).

Or we may write any of the following, were $\lim$ is read as ‘the set of limits of’:

• $x \in \lim \nu$,
• $x \in \lim_i x_i$,
• $x \in \lim x_i$ (suppressing the index $i$ by abuse of notation).

Of course, the right-hand side has a meaning by itself, as the set of limits itself (a subset of the underlying set of $X$, or a subspace of $X$ itself).

If $X$ is a Hausdorff space, then there is at most one point $x$ with the property that the sequence (or net) $\nu$ converges to $x$. Then we may write any of the following, were now $\lim$ is read as ‘the limit of’:

• $x = \lim \nu$,
• $x = \lim_i x_i$,
• $x = \lim x_i$ (suppressing the index $i$ by abuse of notation).

Now the right-hand side by itself is the possibly undefined term for the limit itself (if it exists).

Limits of filters

More generally than sequences, and equivalently to nets, we may speak of limits of filters on $X$. This concept is axiomatized directly in the concept of convergence space. In the case of a topological space $X$, a filter of subsets of $X$ converges to a point $x$ if every neighbourhood of $x$ is contained in the filter.

In the definitions above, equivalent nets (those with equal eventuality filters) always converge to the same point. As every proper filter is the eventuality filter of some net, a proper filter converges to $x$ if any of these nets converges to $x$; the improper filter converges to every point. (In constructive mathematics, we may cover all filters by saying: $F$ converges to $x$ if, on the assumption that $F$ is proper, any of its nets converges to $x$.)

Properties

Relation to limits in the sense of category theory

The limits of category theory are a great generalization of an analogy with the limits discussed here. It turns out, however, that limits in topological spaces (at least) can be viewed as category-theoretic limits. For now, see this math.sx answer.

Examples

• geometric series

Related concepts

• filter

• convergence space

• series

• limit point

• radius of convergence

• uniform convergence, pointwise convergence

• divergent series

• limit of a function

References

• Wikipedia, Limit (mathematics)

Discussion of this history of the concept, with emphasis on its roots all the way back in Zeno's paradoxes of motion is in

• Carl Benjamin Boyer, The history of the Calculus and its conceptual development, Dover 1949

category: analysis