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~

Synopsis:

before~after

The tie character, ~, produces a space between before and after at which the line will not be broken. By default the white space has length 3.33333pt plus 1.66666pt minus 1.11111pt (see Lengths).

Here LaTeX will not break the line between the final two words.

Thanks to Prof.~Lerman.

In addition, despite the period, LaTeX does not use the end-of-sentence spacing (see \@).

Ties prevent the end of line separation of things where that could cause confusion. They also still allow hyphenation (of either of the tied words), so they are generally preferable to putting consecutive words in an \mbox (see \mbox & \makebox).

They are also matters of taste, sometimes alarmingly dogmatic taste, among readers. Nevertheless, here are some usage models, many of them from The TeXbook.

Between an enumerator and its item, such as in references: Chapter~12, or Theorem~\ref{th:Wilsons}, or Figure~\ref{fig:KGraph}.
When cases are enumerated inline: (b)~Show that $f(x)$ is (1)~continuous, and (2)~bounded.
Between a number and its unit: $745.7.8$~watts (the siunitx package has a special facility for this) or 144~eggs. This includes between a month and a date: October~12 or 12~Oct. In general, in any expressions where numbers and abbreviations or symbols are separated by a space: AD~565, or 2:50~pm, or Boeing~747, or 268~Plains Road, or \$$1.4$~billion. Other common choices here are a thin space (see \thinspace & \negthinspace) and no space at all.
When mathematical phrases are rendered in words: equals~$n$, or less than~$\epsilon$, or given~$X$, or modulo~$p^e$ for all large~$n$ (but compare is~$15$ with is $15$~times the height). Between mathematical symbols in apposition with nouns: dimension~$d$ or function~$f(x)$ (but compare with length $l$~or more). When a symbol is a tightly bound object of a preposition: of~$x$, or from $0$ to~$1$, or in common with~$m$.
Between symbols in series: $1$,~$2$, or~$3$ or $1$,~$2$, \ldots,~$n$.
Between a person’s given names and between multiple surnames: Donald~E. Knuth, or Luis~I. Trabb~Pardo, or Charles~XII—but you must give TeX places to break the line so you might do Charles Louis Xavier~Joseph de~la Vall\'ee~Poussin.