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.
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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.
The precise definition depends on what sort of space $X$ is.
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:
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:
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, (quasi)-uniform spaces, pretopological spaces, and (quasi)-uniform convergence spaces.
When one of the conditions above holds, we may write any of the following, where ‘$\to$’ is read as ‘converges to’:
Or we may write any of the following, were $\lim$ is read as ‘the set of limits of’:
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’:
Now the right-hand side by itself is the possibly undefined term for the limit itself (if it exists).
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 $x$.)
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.
Discussion of this history of the concept, with emphasis on its roots all the way back in Zeno's paradoxes of motion is in