In the absence of relevant examples, it is not clear whether Branciari’s concept of weakening the triangle inequality will prove useful in analysis. However, the notion of assigning a ‘distance’ between each two points of an abstract set is fundamental in geometry. According to Blumenthal [[4], p.31], this notion has its origins in the late nineteenth century in axiomatic studies of de Tilly [5]. In his 1928 treatise [6], Karl Menger used the term *halb-metrischer Raume*, *or semimetric space*, to describe the same concept. We begin by summarizing the results of Wilson’s seminal paper [7] on semimetric spaces.

**Definition 1** Let *X* be a set and let
be a mapping satisfying for each
:

I.
, and
;

II.
. Then the pair
is called a *semimetric space*.

In such a space, convergence of sequences is defined in the usual way: A sequence
is said to *converge* to
if
. Also, a sequence is said to be *Cauchy* (or *d*-Cauchy) if for each
there exists
such that
. The space
is said to be *complete* if every Cauchy sequence has a limit.

With such a broad definition of distance, three problems are immediately obvious: (i) *There is nothing to assure that limits are unique* (*thus the space need not be Hausdorff*); (ii) *a convergent sequence need not be a Cauchy sequence*; (iii) *the mapping*
*need not even be continuous*. Therefore it is unlikely there could be an effective topological theory in such a setting.

With the introduction of the triangle inequality, problems (i), (ii), and (iii) are simultaneously eliminated.

VI. (Triangle inequality)

*With*
*X*
*and*
*d*
*as in Definition* 1,

*assume also that for each*
,

**Definition 2** A pair
satisfying Axioms I, II, and VI is called a *metric space*.^{a}

In his study [7], Wilson introduces three axioms in addition to I and II which are weaker than VI. These are the following.

III.

*For each pair of* (

*distinct*)

*points*
,

*there is a number*
*such that for every*
,

IV.

*For each point*
*and each*
,

*there is a number*
*such that if*
satisfies

, then

*for every*
,

V.

*For each*
, there is a number

such that if

satisfy

, then for every

,

Obviously, if Axiom V is strengthened to
, then the space becomes metric. Chittenden [8] has shown (using an equivalent definition) that a semimetric space satisfying Axiom V is always *homeomorphic* to a metric space.

Axiom III is equivalent to the assertion that there do not exist distinct points
and a sequence
such that
as
. Thus, as Wilson observes, the following is self-evident.

**Proposition 1**
*In a semimetric space*, *Axiom *III *is equivalent to the assertion that limits are unique*.

For
, let
. Then Axiom III is also equivalent to the assertion that *X* is Hausdorff in the sense that given any two distinct points
, there exist positive numbers
and
such that
. This suggests the presence of a topology.

**Definition 3** Let
be a semimetric space. Then the distance function *d* is said to be *continuous* if for any sequences
,
and
.

**Remark** Some writers call a space satisfying Axioms I and II a ‘symmetric space’ and reserve the term semimetric space for a symmetric space with a continuous distance function (see, *e.g.*, [9]; *cf.* also [10, 11]). Here we use Menger’s original terminology.

A point *p* in a semimetric space *X* is said to be an *accumulation point* of a subset *E* of *X* if, given any
,
. A subset of a semimetric space is said to be *closed* if it contains each of its accumulation points. A subset of a semimetric space is said to be *open* if its complement is closed. With these definitions, if *X* is a semimetric space with a continuous distance function, then
is an open set for each
and
and, moreover, *X* is a Hausdorff topological space [4].

We now turn to the concept introduced by Branciari.

**Definition 4** ([1])

Let

*X* be a nonempty set, and let

be a mapping such that for all

and all distinct points

, each distinct from

*x* and

*y*:

- (i)
;

- (ii)
;

- (iii)
(quadrilateral inequality).

Then *X* is called a *generalized metric space* (g.m.s.).

**Proposition 2**
*If*
*is a generalized metric space which satisfies Axiom *III, *then the distance function is continuous*.

*Proof* Suppose that

satisfy

and

, where

. Also assume that for

*n* arbitrarily large,

and

. In view of Axiom III, we may also assume that for

*n* sufficiently large,

. Then

Together these inequalities imply

Thus
. □

Therefore if a generalized metric space satisfies Axiom III, it is a Hausdorff topological space. However, the following observation shows that the quadrilateral inequality implies a weaker but useful form of distance continuity. (This is a special case of Proposition 1 of [12].)

**Proposition 3**
*Suppose that*
*is a Cauchy sequence in a generalized metric space*
*X*
*and suppose*
. *Then*
*for all*
. *In particular*,
*does not converge to*
*p*
*if*
.

*Proof* We may assume that

. If

for arbitrarily large

*n*, it must be the case that

. So, we may also assume that

for all

*n*. Also,

for infinitely many

*n*; otherwise, the result is trivial. So, we may assume that

and

for all

with

. Then, by the quadrilateral inequality,

Since

is a Cauchy sequence,

. Therefore, letting

in the above inequalities,

□

We now come to Branciari’s extension of Banach’s contraction mapping theorem. Although in his proof Branciari makes the erroneous assertion that a g.m.s. is a Hausdorff topological space with a neighborhood basis given by

with the aid of Proposition 3, Branciari’s proof carries over with only a minor change. The assertion in [2] that the space needs to be Hausdorff is superfluous, a fact first noted in [12]. See also the example in [13].

**Theorem 1** ([1])

*Let*
*be a complete generalized metric space*, *and suppose that the mapping*
*satisfies*
*for all*
*and fixed*
. *Then*
*f*
*has a unique fixed point*
, *and*
*for each*
.

It is possible to prove this theorem by following the proof given by Branciari up to the point of showing that
is a Cauchy sequence for each
. Then, by completeness of *X*, there exists
such that
. But
, so
. In view of Proposition 3,
.