felix hausdorff

to do without the former of these symbols. In general usage it would ordinarily
be known whether or not a consisted of a single element or a collection of
elements; and such being known, the distinction between the notation aeA and
a — A appears to the reviewer to become unimportant. The notation aeA is
objectionable for the additional reason that the notation X means the set X
plus all of its limiting elements, and to use e in an entirely different sense
is confusing to a reader even though he can soon grasp the correct meaning.
Another confusing thing is the author's use of the term „Entfernung", which
we would translate as „distance", for one thing and the term „Distanz"for a
different thing. The former of these is denoted by xy (for distance from x to
y) and the latter by 'xy. On the whole, however, the system of notation is very
good and is one which authors in the subject would do well to study and follow
wherever it is necessary or desirable to resort to symbolic language. The use
of the term component to replace the rather awkward term maximal connected
subset seems especially desirable to the reviewer.
We consider now §39,1 of the book, which is entitled Bedingungen fiir einfache
Kurven. Here the author lists the following five conditions as necessary
conditions in order that a metric space C be a simple continuous arc.
{a) C is self-compact.
{(3) C contains two points a and h between which it is an irreducible continuum.
(7) C contains two points a and h between which it is irreducibly connected.
{5) [a condition equivalent to (7)].
(e) C is locally connected, i.e., connected im kleinen.
He then proceeds to state (cf. Theorems HI and IV) that the sets of conditions
(a, 7), (Q^,(5), and (a,^, e) are necessary and sufficient in order that C should
be a simple continuous arc. No proof is given for the sufficiency of the sets
(a, 7) and (ce, (S); instead, references are made to N. J. Lennes and W. Sierpinski,
respectively. However, the author gives a complete proof for the sufficiency of
the set of conditions (a, (3, e). This proof is very elementary in nature, and we
do not critizice the author in the least for giving it. It is interesting, however,
to see how the theorem follows by the following line of argument: Since C is
compact, by (ce), and metric and, by (e), is locally connected, it is readily seen
that C satisfies axioms 1, 2, and 4 of R. L. Moore's paper On the foundations of
plane analysis situs.'^ Hence by Theorem 15 of that paper, the proof of which
uses essentially only these axioms, it follows that C is arcwise connected. But,
by (/3), C is an irreducible continuum between some two of its points a and 6;
and since C contains an arc from a to 6, it is clear that C must be identically
this arc.
It is of interest in this connection to note that if C is a subset of a Euclidean
space of any number of dimensions, then the condition a may be omitted from
the set (ce, /^, e), i. e., conditions (/5, e) characterize an arc in an Euclidean space.
For, by (/3), C is an irreducible continuum between some two of its points a and
^Transactions of the American Mathematical Society, vol. 17 (1916), pp. 131-164.
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