A metric is a measurement how far apart each pair elements of a given set are. Without a doubt, a metric is one of the most important notions in mathematics and many other scientific fields. For instance, a metric is used to quantify a dissimilarity (or equivalently similarity) between two objects in some sense. The definition of a metric was proposed by M. Fréchet [4] in 1906.

Definition 1.1. [4] Let Ω be a nonempty set. A function d:Ω×Ω→ℝ+ is called a metric or distance function on Ω if it satisfies the following conditions:

• (1) (identity) d(x,y)=0 if and only if x=y,

• (2) (non-negativity) d(x,y)<0 if x ≠ y,

• (3) (symmetry) d(x,y) = d(y,x) for all x, y ∈ Ω,

• (4) (triangle inequality) d(x,y) ≤ d(x,z) + d(z,y) for all x, y, z ∈ Ω.

The pair (Ω, d) is called a metric space.

In 1963, Gahler [5] generalized an ordinary metric space, called a 2-metric space. It, however, was shown in [6] that not every 2-metric is continuous and there is no strong connection between fixed point theorems in an ordinary metric space and in a 2-metric space, which means that a 2-metric space is not a natural generalization of an ordinary metric space. For this reason, Dhage [2] introduced a newly generalized metric space, called D-metric space, and related fixed point theorems. However, Mustafa and Sims [8] pointed out that similar problems occur in the setting of Dhage, and they [9] proposed an appropriate notion of a generalized metric space. See [1] and references therein for more details.

Definition 1.2. [9] Let Ω be a nonempty set. A function G:Ω×Ω×Ω→ℝ+ is

called a G-metric on Ω if it satisfies the following conditions:

• (G1) G(x,y,z)=0 if x=y=z,

• (G2) G(x,x,y) < 0 for all x,y ∈ Ω with x ≠ y,

• (G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ Ω with y ≠ z,

• (G4) G(x,y,z) = G(x,z,y) = ⋯ (symmetry in all three variables x,y,z),

• (G5) G(x,y,z) ≤ G(x,w,w) + G(w,y,z) for all x,y,z,w ∈ Ω.

The pair (Ω,G) is called a G-metric space. A G-metric space (Ω,G) is said to be symmetric if

• (G6) G(x,y,y) = G(x,x,y) for all x,y ∈ Ω.

More generalized measurement methods are required to be considered in order to analyze more complex data sets such as grouped multivariate data. In this paper, we propose a generalized notion of a metric between n points, called a g-metric. It coincides with the ordinary distance between two points and with the G-metric between three points. Furthermore, we establish fundamental topological notions and properties on the g-metric space including the convergence of sequences and continuity of mappings.

Let ℕ (resp. ℝ) be the set of all nonnegative integers (resp. all real numbers). We denote as ℝ+ the set of all nonnegative real numbers. For a finite set A, we denote the number of distinct elements of A by n(A).

We now propose a new definition of a generalized metric for n number of points instead of two or three points in a given set. For a set Ω, we denote Ωn:=∏ i=1 nΩ.

Definition 2.1. Let Ω be a nonempty set. A function g:Ωn→ℝ+ is called a generalized metric or simply g-metric with dimension n (n≥2) on Ω if it satisfies the following conditions:

• (g1) (positive definiteness) g(x1,…,xn)=0 if and only if x1=⋯=xn,

• (g2) (permutation invariancy) g(x1,…,xn)=g(xσ(1),…,xσ(n)) for any permutation σ on {1,…,n},

• (g3) (monotonicity) g(x1,…,xn)≤g(y1,…,yn) for all (x1,…,xn),(y1,…,yn)∈Ωn with {xi:i=1,…,n}⊊{yi:i=1,…,n},

• (g4) (triangle inequality) for all x1,…,xs,y1,…,yt,w∈Ω with s+t=n

  g(x1,…,xs,y1,…,yt)≤g(x1,…,xs,w,…,w)+g(y1,…,yt,w,…,w).

The pair (Ω, g) is called a g-metric space.

Definition 2.2. A g-metric on Ω is called multiplicity-independent if the following holds

g(x1,…,xn)=g(y1,…,yn)

for all (x1,…,xn),(y1,…,yn)∈Ωn with {xi:i=1,…,n}={yi:i=1,…,n}.

Note that for a given multiplicity-independent g-metric with dimension 3, it holds that g(x,y,y)=g(x,x,y). For a given multiplicity-independent g-metric with dimension 4, it holds that g(x,y,y,y)=g(x,x,y,y)=g(x,x,x,y) and g(x,x,y,z)=g(x,y,y,z)=g(x,y,z,z).

Remark 2.3. If we allow equality under the condition of monotonicity in Definition 2.1, i.e., "g(x1,…,xn)≤g(y1,…,yn) for all (x1,…,xn),(y1,…,yn)∈Ωn with {xi:i=1,…,n}⊆{yi:i=1,…,n}", then every g-metric becomes multiplicity-independent.

Let us explain why the condition (g4) can be considered as a generalization of the triangle inequality. Recall that the triangle inequality condition for a distance function d is d(x,y)≤d(x,z)+d(z,y) for all x,y,z.

The point w is required to measure approximately the distance between x and y with the distances between x and w and between w and y. Note that one cannot measure the distance between x and y by the distances d(x,w₁) and d(y,w₂) with w1≠w2. Consider d(x,y) as a dissimilarity between x and y. Clearly, if x=y, then the dissimilarity is 0, vice versa. Also, the dissimilarity between x and y is same as the dissimilarity between y and x. If x (resp. y) and z (resp. z) are sufficiently similar, then by the triangle inequality x and y must be sufficiently similar.

In the similar way, one can generalize the definition of triangle inequality for the g-metric. Specifically, one can see from the definition of triangle inequality for the g-metric that if both g(x1,…,xs,w,…,w) and g(y1,…,yt,w,…,w) are sufficiently small, then g(x1,…,xs,y1,…,yt) must be sufficiently small. That is, the higher similarities two data sets {x1,…,xs,w} and {y1,…,yt,w} have, the higher similarity data set {x1,…,xs,y1,…,yt} does. Note that w is a necessary point to combine information about similarity for each data set.

The following theorem shows us that g-metrics generalize the notions of ordinary metric and G-metric.

Theorem 2.4. Let Ω be a given nonempty set. The following are true.

• (1) d is a g-metric with dimension 2 on Ω if and only if d is a metric on Ω.

• (2) d is a (resp. multiplicity-independent) g-metric with dimension 3 on Ω if and only if d is a (resp. symmetric) G-metric on Ω.

Remark that since a g-metric with dimension 3 on a nonempty set Ω is a G-metric, any g-metrics with dimension 3 satisfy all properties of the G-metric as shown in [9].

A new g-metric can be constructed from given g-metrics. The proof is left to the reader.

Lemma 2.5. Let (Ω, g) and (Ω,g˜) be g-metric spaces. Then the following functions, denoted by d, are g-metrics on Ω.

• (1) d(x1,x2,…,xn)=g(x1,x2,…,xn)+g˜(x1,x2,…,xn).

• (2) d(x1,x2,…,xn)=ψ(g(x1,x2,…,xn)) where ψ is a function on [0,∞) satisfies

  • (i) ψ is increasing on [0,∞);

  • (ii) ψ(0)=0 ;

  • (iii) ψ(x+y)≤ψ(x)+ψ(y) for all x,y∈[0,∞).

Example 2.6. The following functions, denoted by ψ, satisfy the conditions in Lemma 2.5 (2). Thus, each ψ∘g is a g-metric for any g-metric g.

• (1) (ψ∘g)(x1,…,xn)=kg(x1,…,xn) where ψ(x)=kx with a fixed k>0.

• (2) (ψ∘g)(x1,…,xn)=g(x1,…,xn)1+g(x1,…,xn) where ψ(x)=x1+x.

• (3) (ψ∘g)(x1,…,xn)=g(x1,…,xn) where ψ(x)=x. Furthermore, it is true for ψ(x)=x1/p with a fixed p ≥1.

• (4) (ψ∘g)(x1,…,xn)=log(g(x1,…,xn)+1) where ψ(x)=log(x+1).

• (5) (ψ∘g)(x1,…,xn)=min{k,g(x1,…,xn)} where ψ(x)=min{k,x} with a fixed k>0.

Lemma 2.7. Let g be a g-metric with dimension n on a nonempty set Ω. The following are true:

• (1) g( x,…,x︸s  times,y,…,y)≤g( x,…,x︸s  times,w,…,w)+g( w,…,w︸s  times,y,…,y),

• (2) g(x,y,…,y)≤g(x,w,…,w)+g(w,y,…,y),

• (3) g( x,…,x︸s  times,w,…,w)≤sg(x,w,…,w) and

  g( x,…,x︸s  times,w,…,w)≤(n−s)g(w,x,…,x),

• (4) g(x1,x2,…,xn)≤∑ i=1 ng(xi,w,…,w),

• (5) |g(y,x2,…,xn)−g(w,x2,…,xn)|≤max{g(y,w,…,w),g(w,y,…,y)},

• (6) |g( x,…,x︸s  times,w,…,w)−g( x,…,x︸s˜  times,w,…,w)|≤|s−s˜|g(x,w,…,w).

• (7) g(x,w,…,w)≤(1+(s−1)(n−s))g( x,…,x︸s  times,w,…,w),
  

Proof. (1) and (2) follow from the condition (g4). Note that for a multiplicity-independent g-metric g, it is true that g(y,w,…,w)=g(w,y,…,y).

• (3) By the condition (g4), it follows that

  g( x,…,x︸s  times,w,…,w)≤g( x,…,x︸s−1  times,w,w)+g(x,w,…,w)≤g( x,…,x︸s−2  times,w,w,w)+g(x,w,…,w)+g(x,w,…,w) ⋮≤sg(x,w,…,w).

• (4) By the condition (g2) and (g4), it follows that
  g(x1,x2,…,xn)≤g(x1,w,…,w)+g(x2,x3,…,xn,w)≤g(x1,w,…,w)+g(x2,w,…,w)+g(x3,…,xn,w,w) ⋮≤∑ i=1 ng(xi,w,…,w).
  

• (5) By the condition (g4), we get the inequality

  g(y,x2,…,xn)≤g(w,x2,…,xn)+g(y,w,…,w).

  So

  g(y,x2,…,xn)−g(w,x2,…,xn)≤g(y,w,…,w).

  Similarly, we have

  g(w,x2,…,xn)−g(y,x2,…,xn)≤g(w,y,…,y).

• (6) By (3), it is trivial.

• (7) By Lemma 2.7 (3), we have

  g(x,w,…,w)≤g(x,x,w,…,w)+g(w,x,…,x)≤g(x,x,x,w,…,w)+g(w,x,…,x)+g(w,x,…,x) ⋮≤g( x,…,x︸s  times,w,…,w)+(s−1)g(w,x,…,x)≤g( x,…,x︸s  times,w,…,w)+(s−1)(n−s)g( x,…,x︸s  times,w,…,w)=(1+(s−1)(n−s))g( x,…,x︸s  times,w,…,w).

For a given g-metric, we can construct a distance function.

Proposition 2.8. For any g-metric space (Ω,g), the following are distance functions.

• (1) d(x,y)=g( x,…,x︸s  times,y,…,y)+g( y,…,y︸s  timesx,…,x),

• (2) d(x,y)=g(x,y,…,y)+g(x,x,y,…,y)+⋯+g(x,x,…,x,y),

• (3) d(x,y)=max{g(x1,x2,…,xn):xi∈{x,y}, 1≤i≤n}.

We give several interesting examples of g-metric on a variety of settings in the following.

Example 2.9.

• (1) (Discrete g-metric) For a nonempty set Ω, define d:Ωn→ℝ+ by

  d(x1,…,xn)=0 if x1 =⋯=xn ,1 otherwise 

  for all x1,…,xn∈Ω. Then d is a g-metric on Ω.

• (2) (Diameter g-metric) Define d:ℝ+n→ℝ+ by

  d(x1,…,xn)=max1≤i≤nxi−min1≤j≤nxj

  for all x1,…,xn∈ℝ+. Then d is a g-metric on ℝ+.

• (3) (Average g-metric) For a given metric space (Ω, δ), define d:Ωn→ℝ+ by

  d(x1,…,xn)=1n2∑ i,j=1 nδ(xi,xj)

  for all x1,…,xn∈Ω. Then d is a g-metric on Ω.

• (4) (Max g-metric) For a given metric space (Ω, δ), define d:Ωn→ℝ+ by

  d(x1,…,xn)=max1≤i,j≤nδ(xi,xj)

  for all x1,…,xn∈Ω. Then d is a g-metric on Ω.

• (5) (Shortest path g-metric) For a given metric space (Ω, δ), define d:Ωn→ℝ+ by

  d(x1,…,xn)=minπ∈S∑i=1n−1δ(xπ(i),xπ(i+1))

  for all x1,…,xn∈Ω.

  Here, S denotes the set of all permutations on {1,…,n}. So d(x1,…,xn) is the length of the shortest path connecting x1,…,xn. Finding the shortest path is very important problem in operations research and theoretical computer science, which is also known as the traveling salesman problem[10, 12].
  

• (6) (Smallest ball g-metric) Let Ω be a nonempty subset of ℝn, i.e., Ω can be considered as an n-dimensional data set. Define d:Ωn→ℝ+ by d(x1,…,xn) is the diameter of the smallest closed ball, B, such that {x1,…,xk}⊆B. This is called the smallest enclosing circle problem, which was introduced by Sylvester[11]. For more information, see [3, 7]. It is an open problem that d is a g-metric for any n ≥ 4.

Remark 2.10.

• (1) For a nonempty normed space (Ω,‖⋅‖), let us define a map d:Ωn→ℝ+ by

  d(x1,…,xn)=max1≤i≤n‖xi‖−min1≤j≤n‖xj‖

  for all x1,…,xn∈Ω. Then it is not a g-metric on Ω. In fact, it holds (g2), (g3), and (g4), but does not hold (g1) in general. Indeed, there possibly exist x1,x2,…,xn∈Ω such that ‖x1‖=‖x2‖=⋯=‖xn‖ although xi≠xj for some i≠j.

• (2)In Example 2.9 (3), on a given metric space (Ω,δ)

  d(x1,…,xn)=∑ i,j=1 nδ(xi,xj)

  is a g-metric by Example 2.6 (1). Then this g-metric and the max g-metric in Example 2.9 (4) can be considered as

  d(x1,…,xn)=∑i,j=1 nδ(xi,xj)=||M||1,d(x1,…,xn)=max1≤i,j≤nδ(xi,xj)=||M||∞,

  where M=[mij]1≤i,j≤n is the n×n matrix whose entries are mij=δ(xi,xj). Here, ||⋅||1 and ||⋅||∞ are l1 and l∞ matrix norms, respectively. So it is a natural question whether or not ||M||p for 1<p<∞ is a g-metric on the metric space (Ω,δ).

For a given metric space (Ω,d), we denote the ball centered at x with radius r by Bd(x,r). We define a ball on a g-metric space.

Definition 3.1. Let (Ω,g) be a g-metric space. For x∈Ω and r>0, the ball centered at x with radius r is

Bg(x,r)={y∈Ω:g(x,y,…,y)<r}.

Proposition 3.2. Let (Ω,g) be a g-metric space. Then the following hold.

• (1) If g(x1,x2,…,xn)<r and n({x1,x2,…,xn})≥3, then xi∈Bg(x1,r) for all i=1,…,n.

• (2) If g is multiplicity-independent and g(x1,x2,…,xn)<r, then xi∈Bg(x1,r) for all i=1,…,n.

• (3) Let y∈Bg(x1,r1)∩Bg(x2,r2). Then there exists δ>0 such that Bg(y,δ)⊆Bg(x1,r1)∩Bg(x2,r2).

Proof. Suppose that g(x1,x2,…,xn)<r. Set X={x1,x2,…,xn}.

• (1) Since n(X)≥3, clearly {x1,xi,xi,…,xi}⊊X for each i∈ℕ. By monotonicity of the g-metric, we have g(x1,xi,…,xi)≤g(x1,x2,…,xn)<r. So xi∈Bg(x1,r) for all i∈ℕ.

• (2) It suffices to show that it holds for n(X) = 2. Since a g-metric is multiplicity-independent, g(x1,xi,…,xi)≤g(x1,x2,…,xn)<r.

• (3) Since y∈Bg(x1,r1)∩Bg(x2,r2), it holds that g(xi,y,…,y)<ri for i=1,2. We take δ=min{ri−g(xi,y,…,y):i=1,2}. Then for every z∈Bg(y,δ), by Lemma 2.7 (2) we have g(xi,z,…,z)≤g(xi,y,…,y)+g(y,z,…,z)<g(xi,y,…,y)+δ<ri for each i=1,2. Therefore, Bg(y,δ)⊆Bg(x1,r1)∩Bg(x2,r2).

Due to the preceding proposition, the collection of all balls, B={Bg(x,r):x∈Ω,r>0} forms a basis for a topology on Ω. We call the topology generated by B the g-metric topology on Ω.

Theorem 3.3. Let (Ω,g) be a g-metric space and let d(x,y)=g(x,y,…,y)+g(y,x,…,x). Then

Bg(x1,rn)⊆Bd(x1,r)⊆Bg(x1,r).

Proof. Recall that y∈Bg(x1,r)→g(x1,y,…,y)<r.

(i) Let x∈Bg(x1,rn). Then g(x1,x,…,x)<rn. It follows that

d(x1,x)=g(x1,x,…,x)+g(x,x1,…,x1)≤g(x1,x,…,x)+(n−1)g(x1,x,…,x)≤ng(x1,x,…,x)<r.

So, x∈Bd(x1,r).

(ii) Let x∈Bd(x1,r). Then d(x1,x)=g(x1,x,…,x)+g(x,x1,…,x1)<r.

Since g(x1,x,…,x)≤(n−1)g(x,x1,…,x1), it follows that

nn−1g(x1,x,…,x)≤g(x1,x,…,x)+g(x,x1,…,x1)<r.

Thus, g(x1,x,…,x)<r, i.e., x∈Bg(x1,r) as desired.

Remark 3.4. Every g-metric space is topologically equivalent to a metric space arising from the metric d defined in Theorem 3.3. This makes it possible to transport many concepts and results from metric spaces into the g-metric setting.

Definition 3.5. Let (Ω,g) be a g-metric space. Let x∈ Ω be a point and {xk}⊆Ω be a sequence.

• (1) {xk} converges to x, denoted by {xk}→gx, if for all ε>0 there exists N∈ℕ such that

  i1,…,in−1≥N⇒g(x,xi1,…,xin−1)<ε.

  For such a case, {xk} is said to be convergent in Ω and x is called the limit of {xk}.

• (2) {xk} is said to be Cauchy if for all ε>0 there exists N∈ℕ such that

  i1,…,in≥N⇒g(xi1,…,xin)<ε.

• (3) (Ω,g) is complete if every Cauchy sequence in (Ω,g) is convergent in (Ω,g).

Proposition 3.6. The following are true.

• (1) The limit of a convergent sequence in a g-metric space is unique.

• (2) Every convergent sequence in a g-metric space is a Cauchy sequence.

Proof.Proposition 3.11

• (1) Let (Ω,g) be a g-metric space and let {xk}⊆Ω be a convergent sequence. Suppose that x,y∈Ω are the limits of {xk}. By Definition 3.5 (1), there exists N1,N2∈ℕ such that

  g(x,xi1,…,xin−1)<εn for all i1,…,in≥N1,g(y,xi1,…,xin−1)<εn for all i1,…,in≥N2.

  Set N=max{N1,N2}. If m ≥ N, then by the condition (g4) and Lemma 2.7 (3), it follows that

  g(x,y,y,…,y)≤g(x,xm,xm,…,xm)+g(xm,y,y,…,y)≤g(x,xm,xm,…,xm)+(n−1)g(y,xm,xm,…,xm)<εn+(n−1)εn=ε.

  Since ε is arbitrary, g(x,y,y,…,y)=0. Thus, x=y by the condition (g1).

• (2) Let (Ω,g) be a g-metric space and let {xk}⊆Ω be a convergent sequence with the limit x. By Definition 3.5 (1), there exists N∈ℕ such that

  g(x,xi1,…,xin−1)<εn for all i1,…,in−1≥N.

  By Lemma 2.7 (4) and the monotonicity condition for the g-metric, it follows that

  g(xi1,…,xin)≤∑ k=1ng(xik,x,x,…,x)<∑ k=1nεn=ε.

  Thus, {xk} is a Cauchy sequence in (Ω, g).

Lemma 3.7. Let (Ω,g) be a g-metric space. Let {xk}⊆Ω be a sequence and x∈Ω.

The following are equivalent.

• (1) {xk}→gx.

• (2) For a given ε>0, there exists N∈ℕ such that xk∈Bg(x,ε) for all k≥N.

• (3) limk1,…,ks→∞g( x k 1 ,…,x k s ︸ s  times,x,…,x)=0 for a fixed 1≤s≤n−1. That is, for all ε>0, there exists N∈ℕ such that k1,…,ks≥N implies g(xk1,…,xks,x,…,x)<ε.

Proof. ((1)⇔(2)) It is clear by the definition of convergence.

((2)⇔(3)) Assume that for a given ε>0, there exists N∈ℕ such that k≥N implies xk∈Bg(x,εs), i.e., g(x,xk,…,xk)<εs.

If k1,…,ks≥N, then by Lemma 2.7 (4), we have that g(xk1,…,xks,x,…,x)≤∑ j=1sg(x,xkj,…,xkj)<ε.

((3)⇔(2)) Let ε>0. Assume that there exists N∈ℕ such that

k1,…,ks≥N⇒g(k1,…,ks,x,…,x)<ε(1+(s−1)(n−s)).

If k≥N, then by Lemma 2.7 (7) it follows that

g(x,xk,…,xk)≤(1+(s−1)(n−s))g( x k ,…, x k︸s  times,x,…,x)<ε.

Lemma 3.8. Let (Ω,g) be a g-metric space. Let {xk}⊆Ω be a sequence. The following are equivalent.

• (1) {xk} is Cauchy.

• (2) g(xk,xk+1,xk+1,…,xk+1)→0 as k→∞.

• (3) limk,l→∞g( xk,…,xk︸ s  times,xl,…,xl)=0 for a fixed 1≤s≤n−1.

Proof. ((1)⇒(2)) It is trivial by Definition 3.5 (2).

((2)⇒(3)) Without loss of generality, we can assume k<l. Let ε>0 be given. Then for each m=0,…,l−k−1 there exists Nm∈ℕ such that g(xk+m,xk+m+1,…,xk+m+1)<εn(l−k). Let N=max{N0,…,Nl−k−1}. Then by Lemma 2.7 (3),(4), and the conditions (g4), we have that

g(xk,…,xk︸s  times,xl,…,xl)≤sg(xk,xl,…,xl)         ≤s(g(xk,xk+1,…,xk+1)+g(xk+1,xl,…,xl))          ⋮         ≤s∑i=kl−1g(xi,xi+1,…,xi+1)<ε,

for all k≥N. If k,l≥N, then g( xk,…,xk︸s  timesxl,…,xl)<ε.

((3)⇒(1)) Let ε>0 be given.

Assume that there exists N∈ℕ such that

k,l≥N ⇒g( xk,…,xk︸s  times,xl,…,xl)<εn(1+(s+1)(n−s)).

If i0,i1,…,in≥N, then by Lemma 2.7 (4),(7) it follows that

g(xi0,xi1,…,xin)≤∑ k=0ng(xik,xi0,…,xi0)≤∑ k=0n(1+(s+1)(n−s))g( x i k ,…, x i k ︸s  times,xi0,…,xi0)<ε.

Definition 3.9. Let (Ω,g) be a g-metric space, and let ε>0 be given.

• (1) A set A⊆ Ω is called an ε-net of (Ω,g) if for each x∈ Ω, there exists a ∈ A such that x∈Bg(a,ε). If the set A is finite then A is called a finite ε-net of (Ω,g).

• (2) A g-metric space (Ω,g) is called totally bounded if for every ε>0 there exists a finite ε-net.

• (3) A g-metric space (Ω,g) is called compact if it is complete and totally bounded.

Definition 3.10. Let (Ω1,g1) and (Ω2,g2) be g-metric spaces.

• (1) A mapping T:Ω1→Ω2 is said to be continuous at a point x∈Ω1 provided that for each open ball Bg2(T(x),ε), there exists an open ball Bg1(x,δ) such that T(Bg1(x,δ))⊆Bg2(T(x),ε).

• (2) T:Ω1→Ω2 is said to be continuous if it is continuous at every point of Ω1.

• (3) T:Ω1→Ω2 is called a homeomorphism if T is bijective, and T and T−1 are continuous. In this case, the spaces Ω₁ and Ω₂ are said to be homeomorphic.

• (4) A property P of g-metric spaces is called a topological invariant if P satisfies the condition:

  If a space Ω₁ has the property P and if Ω₁ and Ω₂ are homeomorphic, then Ω₂ also has the property P.

Proposition 3.11. Let (Ω1,g1) and (Ω2,g2) be g-metric spaces, and let T:Ω1→Ω2 be a mapping. Then the following are equivalent.

• (1) T is continuous.

• (2) For each point x∈Ω1 and for each sequence {xk} in Ω₁ converging to x,{T(xk)} converges to T(x).

Proof. ((1)⇒(2)) Let x∈Ω1, and let {xk} be a sequence in Ω1 converging to x. Since T:Ω1→Ω2 is continuous, for a given ε>0 there exists δ>0 such that T(Bg1(x,δ))⊆Bg2(T(x),ε(n−1)−2). Since {xk}→gx, there is N∈ℕ such that g(x,xi1,…,xin−1)<δ for all i1,…,in−1≥N. Thus g(x,xik,…,xik)<δ for each k=1,…,n−1. Then the continuity of T gives rise to the inequality

g(T(x),T(xik),…,T(xik))<ε(n−1)2

for each k∈ℕ. By Lemma 2.7 (3) and (4) we have

g(T(x),T(xi1),…,T(xin−1))≤∑ k=1 n−1g(T(xik),T(x),…,T(x))≤∑ k=1 n−1(n−1)g(T(x),T(xik),…,T(xik))<ε.

Therefore, {T(xk)} converges to T(x).

((2)⇒(1)) Suppose that T is not continuous, i.e. there exists x∈Ω1 such that T is not continuous at x. Then there exists ε>0 such that for each δ > 0 there is y∈Ω1 with g(x,y,…,y)<δ but g(T(x),T(y),…,T(y))≥ε. Then for each k∈ℕ we can take xk∈Ω1 such that g(x,xk,…,xk)<1k but g(T(x),T(xk),…,T(xk))≥ε. Hence, {xk} converges to x but {T(xk)} does not converges to T(x), which contradicts to (2).