It is a classical question in topology, that what kind of topological spaces can be embedded into finite dimensional Euclidean spaces (endowed with the usual Euclidean topology). To motivate the present paper, first we recall some well known results in related investigations.
A classical characterization theorem can be recalled as follows. The covering dimension of is finite if and only if there exists such that for every open covering C of there exists an open covering that refines C and each point of X belongs to at most elements of (the smallest n satisfying this property is defined to be the dimension of ; if there are no such n, then is defined to be infinite dimensional). Clearly, if can be embedded into a finite dimensional Euclidean space, then is metrizable. By a classical theorem, for a compact, metrizable space the followings are equivalent:
(1) is finite dimensional (say, its dimension is n);
(2) is homeomorphic to a subspace of a finite dimensional (more concretely, -dimensional) Euclidean space
(for a proof we refer to e.g. Corollary 50.9 of  ). Recall, that for a positive real number , a function is defined to be an -map if and only if for all the diameter of is at most . Thus, if is “small” then an -map is “almost injective”. The usual proof for the harder direction of the above equivalence is based on showing that if the dimension of is , then for all , the set of continuous -mappings of into is dense and open; then the Baire category theorem implies the existence of a continuous embedding of into .
Further, as explained in Theorem 2.2 of  , for a compact, metrizable space , the above (1) and (2) are also equivalent with
(3) for all positive , admits an -map to an n-dimensional simplicial complex.
In fact, Theorem 2.2 of  provides several statements equivalent with the above (1), (2) and (3) and recalls further notions of dimensions of topological spaces. All of these dimensions are equivalent for compact metric spaces. For further details on related notions and results we refer to the rather comprehensive survey article  and to the references therein.
Recall e.g. from  , that the metric space is defined to be totally bounded if and only if for all positive there exists a finite collection of -balls of that covers X. Further, is compact if and only if it is totally bounded and complete (that is, every Cauchy sequence of is convergent).
The main result of the present paper is Theorem 10 where we show, that
(i) if is totally bounded, then for all , there exists a continuous -mapping of into a finite dimensional Euclidean space , further
(ii) estimations for n are also provided in terms of and structural properties of .
We should make the following remarks: item (i) above is well known. Thus, “almost injective” functions still exist, even if is infinite dimensional—of course, if is infinite dimensional, then n depends on . The known proofs are existential, they cannot provide any upper bound for n. However our proof for (i) is effective in the sense, that (as stated in (ii) above), based on it, one can establish estimations on n in terms of and structural properties of .
The structure of the paper is rather simple: we close this section by fixing our notation and Section 2 contains the proofs. As we mentioned, our main goal is to prove Theorem 10.
Our notation is mostly standard, but the following list may help.
Throughout denotes the set of natural numbers. In addition, and denotes the set of real numbers, and the set of positive real numbers, respectively.
Let be a metric space, and let be a non-negative real number. As usual, the open -ball at a is the set
Definition 1 A family of -balls is defined to be a -net if and only if it covers X, that is,
Thus, is a completely bounded metric space if and only if for all positive there exists a finite -net in . In addition, as it is well known, is compact if and only if it’s metric is totally bounded and complete (i.e. every Cauchy sequence is convergent in ). For further details we refer to  , as well.
If is a completely bounded metric space, then denotes the smallest cardinality for which there exists a -sized -net of .
Definition 2 Let be a metric space, let and let . Then the -type of b over A in is defined to be the function , such that for all we have
Thus, the -type of b is just the function describing the distances of b from elements of A.
By a -type over A we mean a function which is of the form for some .
Keeping the notation introduced so far, we say, that realizes the -type p in if and only if .
Remark 3 We motivate and explain the above terminology as follows. Let be a metric space. One can associate a relational structure to in the following way. If d is a distance of , that is, then the binary relation is defined to be
Thus, the relational structure completely describes and, at the same time, it can be treated as a model for an appropriate first order language. Then our definition of -types is essentially the same, as atomic types of the first order relational structure in the usual, model theoretic sense.
Lemma 1 Let be a metric space, let be finite and let . Then for the function , we have
particularly, f is continuous.
Proof. Enumerate and for each let , . By a slight abuse of notation, we have
for all .
Let be fixed in this paragraph. Then for any the triangle inequality yields
so we have
Similarly, interchanging x and y in the previous estimations, we obtain
Consequently, for all we have
The following notion is an approximate version of splitting introduced in  .
Definition 4 Let be a metric space, let , let p be a -type over A in and let be non-negative real numbers. Then we say, that p is -splitting over B if and only if there exist such that for all we have
but whenever a realizes p, we have
Keeping the notation introduced in the above definition, intuitively is -splitting over B if and only if there exist such that and are “indiscernible from the viewpoint of B modulo ”, but a “distinguishes them modulo ”.
The following theorems will be essential in this paper. Some variants of them (in different contexts) had been utilized e.g. in  ,  and in  .
Theorem 5 Let be a totally bounded metric space, let ,
let and let be arbitrary. Suppose
is a strictly increasing sequence of subsets of such that for all the type is -splitting over . Then .
Proof. Let be a -net of with smallest possible cardinality. By our assumption on splitting, for each there exist such that for all we have
Assume, seeking a contradiction, that . By the pigeonhole principle, there exists such that (the -ball in our net) contains at least two 's; more precisely, there exist with . Since is a -ball, it follows, that . Therefore, by the triangle inequality,
and by symmetry,
It follows, that
It follows, that
Combining these, we get
Since , this contradicts to (15) above, and the proof is complete.
Definition 6 Let be a metric space, let and let . Then is defined to be an -basis for a if and only if for any with , the type
does not -split over A.
1) Clearly, if A is an -basis for and , then A is a -basis for a, as well.
2) In addition, if A is an -basis for and then B is an -basis for , as well.
Theorem 8 Let be a totally bounded metric space, let and let . Then there exist and such that is an -basis for a.
In fact, arbitrary is suitable and can be chosen so, that is satisfied, as well.
Proof. Let be an arbitrary positive real number.
Suppose, seeking a contradiction, that the consequence of the theorem is not true. By recursion, we define finite subsets for every natural number n, such that the following stipulations are satisfied:
(i) , in fact, ;
(ii) is -splitting over .
Let and suppose has already been defined for all such that stipulations (i) and (ii) are satisfied. Then, by our indirect assumption, there exists with such that is -splitting over . This means, that there exist such that for all we have
Let . Then stipulations (i), (ii) remain true. In this way, one can define for all ; this contradicts to Theorem 5. Thus, the proof is complete: can be chosen to be some (note, that an inspection shows, that each has cardinality at most 2n and ).
Lemma 2 Let be a totally bounded metric space, let and let . If A is an -basis for and , moreover and are pairwise distinct and for all , then A is an -basis for b.
Proof. Assume, seeking a contradiction, that A is an -basis for and , but not an -basis for b. Then there exist such that for all we have
By assumption, and are pairwise distinct. It follows, that there exists such that , that is, .
Combining these estimations, we get
Completely similarly, one also can conclude, that
Observe moreover, that because A is an -basis for . But then,
contradicting to (28).
Theorem 9 Let be a totally bounded metric space. Then for each
there exists a finite set such that A is an -basis for all .
In fact, we have .
Proof. Let be an -net of with . Let
and let . Enumerate as For each there exist pairwise different . By Theorem 8, for each and there exists an -basis for with . Similarly, again by Theorem 8, for each and for each there exists an -basis for a with . Finally, let
We claim, that A satisfies the conclusion of the theorem. By construction, clearly
as desired. In addition, let be arbitrary. Then there exists such that .
If then by construction, A contains an -basis for b, which is also an -basis for b because of Remark 7 (1). Hence, by Remark 7 (2), A is an -basis for b.
If , then for some . By construction, for all we
have . Therefore, by the triangle inequality, for all we have
Hence, by Lemma 2, is an -basis for b. Since , it follows from Remark 7 (2), that A is an -basis for
b, as well. Since b was arbitrary, the proof is complete.
Lemma 3 Let be a metric space and let be arbitrary.
Assume is an -basis for all . If is not isolated,
then for all
Proof. Since x is not an isolated point of , there exists . By assumption, A is an -basis for z, hence
But , so it follows, that . Hence, by the triangle inequality,
Let be a metric space, let and let Y be any set. As we mentioned in the introduction, according to the terminology of e.g.  , a function is defined to be an -map if and only if for all , the diameter of is at most , or equivalently,
Now we are able to state and prove the main result of the paper: we give a new proof for the fact, that each totally bounded metric space admits a continuous -map into some finite dimensional Euclidean space (endowed with the usual Euclidean metric). Further, we provide upper bounds for n.
Theorem 10 Let be a totally bounded metric space, let
and let .
1) if does not contain isolated points, then there exist and an -map such that, for all we have , particularly f is continuous.
2) if has countably many isolated points, then there exist and a continuous -map .
3) if is compact, then there exist and a continuous -map .
Proof. First we show (1). By Theorem 9, there exists such that A is an
-basis for all and . Let and let ,
for all . Then, by Lemma 1, for all we have
Further, by assumption, does not contain isolated points. Hence, by Lemma 3 f is an -map, as desired.
To show (2), enumerate all the isolated points of X as . As in (1),
by Theorem 9 there exists such that A is an -basis for all
and . Let and let ,
for all . Then, by Lemma 1, for all we have
Let , let be the function
and let ,
First we show, that f is continuous. To do so, let and let be arbitrary. We shall show, that there exists such that for all we have .
Case 1: x is isolated in . Then there exists such that , hence, for all we have .
Case 2: x is not isolated in . Then, by 45, there exists such that for all we have . In addition, there exists
with . Choose such that if then
and finally let . Now let .
If y is not isolated, then
If y is isolated, then for some and hence
thus, f is continuous, as desired.
Next we show, that f is an -map. To do so, assume are such, that . Observe, that by construction, for any , the isolated point is the unique element such that the last coordinate of
is equal to . Hence neither x nor y are isolated. Therefore
hence . Combining this with the definition of , we obtain . But then, Lemma 3 implies .
To show (3), we note, that (e.g. by Corollary 4.1.16 of  ) a compact metric space is second countable, hence it may contain countably many isolated points, only. Thus, (2) implies (3).
3. Concluding Remarks
In Theorem 10 we have given a new proof for the fact, that each totally bounded metric space admits a continuous -map into some finite dimensional Euclidean space (endowed with the usual Euclidean metric). Further, we provided upper bounds for n in terms of and structural properties of . Our proof had been obtained as follows:
• As recalled in Remark 3, there is a well known method that associates a first order structure to a metric space ;
• if is totally bounded, then has nice properties inspired by (model theoretic) stability theory, more concretely, as shown in Theorem 5, if
, then each increasing chain of -splitting -types in
should have finite length;
• as shown in Theorem 10, using the above observation one can construct an appropriate continuous -map from to a finite dimensional Euclidean space.
The second item of the above list motivates the following problem which may serve as a starting point of a further investigation.
Open problem 11 Is the converse of Theorem 5 true, as well? That is, if is a metric space such that, for all the length of any increasing chain of
-splitting -types in is finite, then does it follow, that is
This work has been supported by Hungarian National Foundation for Scientific Research grant K129211.