0 Introduction and preliminaries

In 1960, nonstandard analysis was founded by American mathematical logician ROBINSON^[1]. As a new branch of mathematics and new mathematical method, nonstandard analysis was a new mathematical theory that used nonstandard models to study various mathematical problems. In nonstandard analysis, nonstandard topology was an important reaching field.

As usual, there were two ways to study nonstandard topology. One was to research the general topological space^[1-2] or uniform space^[3-4] by nonstandard analysis methods, the other was directly to study the topology on a nonstandard set^[5-6]. In this paper, the latter will be shown. On two nonstandard sets, Fin(^*X) and ^*X, *-topology $ \mathscr{T} $^* and s-topology $ \mathscr{T} $^s induced by metric space (X, ρ) are studied in polysaturated nonstandard model. In *-topological space (Fin(^*X), $ \mathscr{T} $^*), the following conclusions are proved: (1)Every internal set is compact; (2) Every open set is saturated; (3) Standard part mapping st is continuous. In s-topological space (^*X, $ \mathscr{T} $^s), the relations, between in (X, $ \mathscr{T} $) and in (^*X, $ \mathscr{T} $^*), of closure operators and interior operators are discussed, respectively. And it is obtained that mapping * between two categories is a functor.

Firstly, some basic concepts and conclusions are recalled. For details, [1, 2, 5, 7] can be referenced. In this paper, X is any non-empty set. N, R and R⁺ denote the sets of the natural, real and positive real numbers, respectively.

Definition 1  The mapping ρ : X × X → R is called a metric on X if for any x, y, z ∈ X,

(1) ρ(x, y) ≥ 0, and ρ(x, y) = 0 if and only if x = y;

(2) ρ(x, y) = ρ(y, x);

(3) ρ(x, y) ≤ ρ(x, z) + ρ(z, y).

The pair (X, ρ) is called a metric space.

For each x ∈ X and r ∈ R⁺, B(x, r) = {y ∈ X | ρ(x, y) < r} is called an open ball in metric space (X, ρ) with center x and radius r. The collection of all open balls is denoted by B, i.e.

$ \mathit{\boldsymbol{B}} = \left\{ {B\left( {x, r} \right)|x \in X, r \in {\boldsymbol{\rm{R}}^ + }} \right\}. $

As we all known, $ \mathscr{B} $ is a topological base on X. The topology generated by $ \mathscr{B} $ is denoted as $ \mathscr{T} $.

About nonstandard analysis, let S be an individual set. The superstructure V(S) can be inductively defined as follows:

$ \begin{array}{l} {V_0}\left( S \right) = S, \;\;{V_{n + 1}}\left( S \right) = {V_n}\left( S \right) \cup {\mathscr{P}}\left( {{V_n}\left( S \right)} \right), \;\;n \in \boldsymbol{\rm{N}}, \\ V\left( S \right) = \mathop \cup \limits_{n \in \boldsymbol{\rm{N}}} {V_n}\left( S \right) = {V_0}\left( S \right) \cup {V_1}\left( S \right) \cup {V_2}\left( S \right) \cup \cdots , \end{array} $

where $ \mathscr{P} $(V_n(S)) is the power set of V_n(S), i.e. the collection of all subsets of V_n(S).

Suppose that X ∪ R ∈ V(S). It can be easily obtained that mathematical objects used in this paper are all in superstructure V(S), such as open sets, topology, metric, functions on X etc. Given that V(^*S) is polysaturated nonstandard model of V(S).

Lemma 1  (Saturation principle)^[6] V(^*S) is polysaturated nonstandard model of V(S) if and only if for every internal family of sets with the finite intersection property has a non-empty intersection.

1 *-Topology $ \mathscr{T} $^* on Fin(^*X)

In this section, a kind of topology called *-topology on Fin(^*X) will be constructed, and some properties about it will be discussed.

By transfer principle, mapping ^*ρ : ^*X × ^*X → ^*R satisfies conditions (1)~(3) in definition 1. Since the value of ^*ρ is in hyper-real ^*R, the nonstandard extension (^*X, ^*ρ) is a hyper-metric space.

Definition 2  Let (X, ρ) be a metric space.

(1) A point a ∈ ^*X is said to be finite in (^*X, ^*ρ) if ^*ρ(x, a) is finite for some x ∈ X. And Fin(^*X) denotes the set of all finite points in (^*X, ^*ρ).

(2) A point a ∈ ^*X is said to be near-standard in (^*X, ^*ρ) if ^*ρ(x, a) is infinitesimal for some x ∈ X. And ns(^*X) denotes the set of all near-standard points in (^*X, ^*ρ).

Since 0 is infinitesimal and infinitesimal is finite, every standard point is near-standard point and near-standard point is finite point. Thus, X ⊆ ns(^*X) ⊆ Fin(^*X) ⊆ ^*X.

Consider the copy $ \mathscr{B} $ of collection B of all open balls,

$ \mathscr{B} = \left\{ {^ * B\left( {x, r} \right)|B\left( {x, r} \right) \in \mathit{\boldsymbol{B}}} \right\}; $

Since

∪ $ \mathscr{B} $ = ∪{^*B(x, r) | B(x, r) ∈ B} = Fin(^*X), $ \mathscr{B} $ can be a subbase for a topology on Fin(^*X).

Definition 3  The topology generated by $ \mathscr{B} $ is called *-topology on Fin(^*X), and denoted by $ \mathscr{T} $^*.

Theorem 1  Let (X, ρ) be a metric space. Then

(1) For every internal subset A ⊆ Fin(^*X), A is compact in Fin(^*X) under $ \mathscr{T} $^*.

(2) X is dense in Fin(^*X) under $ \mathscr{T} $^*.

Proof  (1) Let A ⊆ Fin(^*X) be internal and {G_λ ∈ $ \mathscr{T} $^* | λ ∈ Λ} be arbitrary $ \mathscr{T} $^*-open cover of A. Then A ⊆ $ \mathop \cup \limits_{\lambda \in \mathit{\Lambda }} $ G_λ. For each a ∈ A, there is λ_a ∈ Λ such that a ∈ G_λa. By definition of $ \mathscr{T} $^*, there exists x_a ∈ X and r_a ∈ R⁺ such that a ∈ ^*B(x_a, r_a) ⊆ G_λa. Since A ⊆ $ \mathop \cup \limits_{a \in A} $^*B(x_a, r_a), A-$ \mathop \cup \limits_{a \in A} $^*B(x_a, r_a)=$ \mathop \cap \limits_{a \in A} $(A-^*B(x_a, r_a))=Ø. By Saturation principle(lemma 1), there are finite elements ^*B(x_i, r_i) (i = 1, 2, …, n) in {^*B(x_a, r_a) | a ∈ A} such that

$ \mathop \cap \limits_{i = 1}^n \left( {A - {}^ * B\left( {{x_i},{r_i}} \right)} \right) = A - \mathop \cup \limits_{i = 1}^n {}^ * B\left( {{x_i},{r_i}} \right) = \emptyset , $

i.e. A ⊆ $ \mathop \cup \limits_{i = 1}^n $^*B(x_i, r_i). Therefore, there is a finite subcover {G_i| i = 1, 2, …, n} of A in {G_λ | λ ∈ Λ}, where G_i ∈{G_λa| a ∈ A} (i = 1, 2, …, n) is corresponding to ^*B(x_i, r_i). Thus, internal set A is compact in Fin (^*X) under $ \mathscr{T} $^*.

(2) Let G be arbitrary non-empty $ \mathscr{T} $^*-open set in Fin(^*X). For each a ∈ G, by definition of $ \mathscr{T} $^*, there exist x ∈ X and r ∈ R⁺ such that a ∈ ^*B(x, r) ⊆ G. So, there is x ∈ X such that x ∈ G, i.e. X ∩ G ≠ Ø for every non-empty G ∈ $ \mathscr{T} $^*. The proof is finished.

Definition 4  Let (X, ρ) be a metric space. For each a ∈ ^*X, the set

$ m\left( a \right) = \cap \left\{ {^ * B\left( {x, r} \right) \in \mathscr{B}|a{ \in ^ * }B\left( {x, r} \right)} \right\} $

is called the monad of point a.

A subset A ⊆ ^*X is said to be saturated if m(a) ⊆ A for all a ∈ A.

It can be easily to see that

(1) m(a) = ∩ {^*G | a ∈ ^*G, G ∈ $ \mathscr{T} $} = {b ∈ ^*X |^*ρ(a, b) is infinitesimal}.

(2) A is saturated if and only if

$ A = \cup \left\{ {m\left( a \right)|a \in A} \right\}. $

Theorem 2   Let (X, ρ) be a metric space. G is saturated for every G ∈ $ \mathscr{T} $^*.

Proof  Let G ∈ $ \mathscr{T} $^*. For each a ∈ G, there are x ∈ X and r ∈ R⁺ such that a ∈ ^*B(x, r) ⊆ G. Since m(a) ⊆ ^*B(x, r), m(a) ⊆ G. Hence, G is saturated.

Define mapping st : ns(^*X) → X, for each a ∈ ns(^*X), st(a) = x if and only if ^*ρ(x, a) is infinitesimal.

As we all known, metric space is also Hausdorff space. For metric space (X, ρ), mapping st is well-defined. It is clear that st(a) = x if and only if a ∈ m(x). Generally, for any A ⊆ ns(^*X), define

$ st\left[ A \right] = \left\{ {x \in X|st\left( a \right) = x, a \in A} \right\}. $

If A = {a}, st({a}) is also denoted as st(a).

Definition 5  Mapping st defined above is called standard part mapping.

Lemma 2^[6]  Let (X, ρ) be a metric space. For any A ⊂ X, cl_X (A) = st[^*A], where cl_X (A) is the closure of A in X under $ \mathscr{T} $.

Theorem 3  Standard part mapping st : ns(^*X)→ X is continuous with respect to $ \mathscr{T} $^* and $ \mathscr{T} $.

Proof  For each a ∈ ns(^*X), let st(a) = x ∈ X, G, H ∈ $ \mathscr{T} $ such that x ∈ G ∩ H and cl_X (G) ⊆ H.Then m(x) ⊆ ^*G. Hence ^*G is a $ \mathscr{T} $^*-open neighborhood of a in ns(^*X). By lemma 2, x = st(a) ∈ st[^*G]=cl_X (G) ⊆ H. Therefore, mapping st is continuous.

2 s-Topology $ \mathscr{T} $^s on ^*X

At last, s-topology $ \mathscr{T} $^s will be constructed on ^*X, where (X, ρ) is a metric space. Some properties will be shown.

Let (X, ρ) be a metric space. Consider the following copy T of topology $ \mathscr{T} $ generated by $ \mathscr{B} $, which is the collection of all open balls in (X, ρ).

$ \mathit{\boldsymbol{T}} = \left\{ {^ * G|G \in \mathscr{T}} \right\}, $

although T is not a topology, it forms a base for a topology on ^*X since ^*X ∈ T.

Definition 6  The topology generated by T is called s-topology on ^*X, and denoted by $ \mathscr{T} $^s.

Now, three topological spaces have been obtained, (X, $ \mathscr{T} $), (Fin(^*X), $ \mathscr{T} $^*) and (^*X, $ \mathscr{T} $^s) are all induced by metric space (X, ρ). Since $ \mathscr{B} $ is base of $ \mathscr{T} $, it is easily to see that

(X, $ \mathscr{T} $) ≤ (Fin(^*X), $ \mathscr{T} $^*) ≤ (^*X, $ \mathscr{T} $^s),

which relation ≤ is partial order relation between topological subspace.

Theorem 4  Let (X, ρ) be a metric space.

(1) Every internal subset A ⊆ ^*X is compact under $ \mathscr{T} $^s.

(2) X is dense in ^*X under $ \mathscr{T} $^s.

Proof  The proof is similar to theorem 1.

Theorem 5  For every A ⊆ X,

(1) ^*(cl_X (A)) = cl_^*X(^*A), where cl_X and cl_^*X are the closure operators in (X, $ \mathscr{T} $) and(^*X, $ \mathscr{T} $^s), respectively.

(2) ^*(int_X (A)) = int_^*X(^*A), where int_X and int_^*X are the interior operators in (X, $ \mathscr{T} $) and(^*X, $ \mathscr{T} $^s), respectively.

Proof  Only (2) is proved.

$ \begin{array}{l} ^ * \left( {{{{\mathop{\rm int}} }_X}\left( A \right)} \right)\;\; \supseteq \cup \left\{ {^ * G{|^ * }\left( {{{{\mathop{\rm int}} }_X}\left( A \right)} \right){ \supseteq ^ * }G, G \in \mathscr{T}} \right\} = \\ \;\;\;\;\; \cup \left\{ {^ * G|{{{\mathop{\rm int}} }_X}\left( A \right) \supseteq G, G \in \mathscr{T}} \right\} = \\ \;\;\;\;\; \cup \left\{ {^ * G|A \supseteq G, G \in \mathscr{T}} \right\} = \cup \left\{ {^ * G{|^ * }A} \right. \supseteq \\ \;\;\;\;\;\left. {^ * G, G \in \mathscr{T}} \right\} = {{\mathop{\rm int}} _{^ * X}}\left( {^ * A} \right). \end{array} $

Conversely, by transfer principle, A ⊇ int_X (A) implies ^*A ⊇ ^*(int_X (A)). Since int_X (A) ∈ $ \mathscr{T} $ implies ^*(int_X (A)) ∈ $ \mathscr{T} $^s, int_^*X (^*A) ⊇ ^*(int_X (A)). The proof is finished.

In the following conclusion, the mapping * will be a functor between two categories $ \mathscr{C} $ and $ \mathscr{C} $^s.

Let (X, ρ) be a metric space. The topologies $ \mathscr{T} $ and $ \mathscr{T} $^s are generated by B and T, respectively. Two categories $ \mathscr{C} $ and $ \mathscr{C} $^s are defined as follows:

(1) The objects of $ \mathscr{C} $ and $ \mathscr{C} $^s are X and ^*X, respectively.

(2) The homomorphism of $ \mathscr{C} $ and $ \mathscr{C} $^s are continuous functions on topological spaces (X, $ \mathscr{T} $) and on (^*X, $ \mathscr{T} $^s), respectively.

Theorem 6  Let $ \mathscr{C} $ and $ \mathscr{C} $^s be two categories defined above. Mapping * : $ \mathscr{C} $→$ \mathscr{C} $^s is a functor.

Proof Step 1  Clearly, for each x ∈ X, ^*(x)=^*x ∈^*X.

Step 2  ^*f is a function on ^*X for every function f on X. In fact, since f is a function on X, the following sentence holds:

$ \begin{array}{l} | = \left[ {\forall x \in X} \right]\left[ {\exists y \in X} \right]\left[ {\left[ {\left\langle {x, y} \right\rangle \in f} \right]} \right. \wedge \\ \;\;\;\;\;\;\left. {\left[ {\forall z \in X} \right]\left[ {\left[ {\left\langle {x, z} \right\rangle \in f} \right] \to \left[ {z = y} \right]} \right]} \right]. \end{array} $

By transfer principle,

$ \begin{array}{l} | = \left[ {\forall x{ \in ^ * }X} \right]\left[ {\exists y{ \in ^ * }X} \right]\left[ {\left[ {\left\langle {x, y} \right\rangle { \in ^ * }f} \right]} \right. \wedge \\ \;\;\;\;\;\;\left. {\left[ {\forall z{ \in ^ * }X} \right]\left[ {\left[ {\left\langle {x, z} \right\rangle { \in ^ * }f} \right] \to \left[ {z = y} \right]} \right]} \right]. \end{array} $

That is, ^*f is a function on ^*X.

Step 3  Let function f be continuous on (X, $ \mathscr{T} $). For any G ∈ $ \mathscr{T} $, ^*G ∈ $ \mathscr{T} $. Then ^*f^-1 [^*G] = ^*(f^-1 [G]) ∈ $ \mathscr{T} $. Function ^*f : (^*X, $ \mathscr{T} $^s) → (^*Y, $ \mathscr{T} $^s) is continuous, since $ \mathscr{T} $ are bases for $ \mathscr{T} $^s.

Step 4  For any continuous functions f and g on X, by step 3, ^*f, ^*g and ^*(f о g) are all continuous on ^*X, since f о g is continuous on X. Then by definition of f о g, the following sentence holds:

$ \begin{array}{l} | = \left[ {\forall x \in X} \right]\left[ {\exists y \in X} \right]\left[ {\left[ {\left\langle {x, y} \right\rangle \in f \circ g} \right]} \right. \leftrightarrow \\ \;\;\;\;\;\;\left. {\left[ {\exists z \in X} \right]\left[ {\left[ {\left\langle {x, z} \right\rangle \in g} \right] \wedge \left[ {\left\langle {z, y} \right\rangle \in f} \right]} \right]} \right]. \end{array} $

By transfer principle,

$ \begin{array}{l} | = \left[ {\forall x{ \in ^ * }X} \right]\left[ {\exists y{ \in ^ * }X} \right]\left[ {\left[ {\left\langle {x, y} \right\rangle } \right.} \right. \in \\ \left. {^ * f \circ g} \right] \leftrightarrow \left[ {\exists z{ \in ^ * }X} \right]\left[ {\left[ {\left\langle {x, z} \right\rangle { \in ^ * }g} \right] \wedge } \right.\\ \left. {\left. {\left[ {\left\langle {z, y} \right\rangle { \in ^ * }f} \right]} \right]} \right]. \end{array} $

That is, ^* (f о g) = ^*f о ^*g.

Step 5  For identity 1_X on X, the following sentence holds:

$ | = \left[ {\forall x \in X} \right]\left[ {\left\langle {x, x} \right\rangle \in {1_X}} \right]. $

By transfer principle,

$ | = \left[ {\forall x{ \in ^ * }X} \right]\left[ {\left\langle {x, x} \right\rangle { \in ^ * }\left( {{1_X}} \right)} \right]. $

That is, ^*(1_X) = 1_^*X.

So, mapping * : $ \mathscr{C} $→$ \mathscr{C} $^s is a functor.