Subspaces of Metric Spaces

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Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

Definition

Given a topological space and a subset of , the subspace topology on is defined by

That is, a subset of is open in the subspace topology if and only if it is the intersection of with an open set in . If is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of . Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

Alternatively we can define the subspace topology for a subset of as the coarsest topology for which the inclusion map

is continuous.

More generally, suppose is an injection from a set to a topological space . Then the subspace topology on is defined as the coarsest topology for which is continuous. The open sets in this topology are precisely the ones of the form for open in . is then homeomorphic to its image in (also with the subspace topology) and is called a topological embedding.

A subspace is called an open subspace if the injection is an open map, i.e., if the forward image of an open set of is open in . Likewise it is called a closed subspace if the injection is a closed map.

Terminology

The distinction between a set and a topological space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. Thus, whenever is a subset of , and is a topological space, then the unadorned symbols "" and "" can often be used to refer both to and considered as two subsets of , and also to and as the topological spaces, related as discussed above. So phrases such as " an open subspace of " are used to mean that is an open subspace of , in the sense used below; that is: (i) ; and (ii) is considered to be endowed with the subspace topology.

Examples

In the following, represents the real numbers with their usual topology.

  • The subspace topology of the natural numbers, as a subspace of , is the discrete topology.
  • The rational numbers considered as a subspace of do not have the discrete topology ({0} for example is not an open set in ). If a and b are rational, then the intervals (a, b) and [a, b] are respectively open and closed, but if a and b are irrational, then the set of all rational x with a < x < b is both open and closed.
  • The set [0,1] as a subspace of is both open and closed, whereas as a subset of it is only closed.
  • As a subspace of , [0, 1] ∪ [2, 3] is composed of two disjoint open subsets (which happen also to be closed), and is therefore a disconnected space.
  • Let S = [0, 1) be a subspace of the real line . Then [0, ) is open in S but not in . Likewise [, 1) is closed in S but not in . S is both open and closed as a subset of itself but not as a subset of .

Properties

The subspace topology has the following characteristic property. Let be a subspace of and let be the inclusion map. Then for any topological space a map is continuous if and only if the composite map is continuous.

Characteristic property of the subspace topology

This property is characteristic in the sense that it can be used to define the subspace topology on .

We list some further properties of the subspace topology. In the following let be a subspace of .

  • If is continuous then the restriction to is continuous.
  • If is continuous then is continuous.
  • The closed sets in are precisely the intersections of with closed sets in .
  • If is a subspace of then is also a subspace of with the same topology. In other words the subspace topology that inherits from is the same as the one it inherits from .
  • Suppose is an open subspace of (so ). Then a subset of is open in if and only if it is open in .
  • Suppose is a closed subspace of (so ). Then a subset of is closed in if and only if it is closed in .
  • If is a basis for then is a basis for .
  • The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.

Preservation of topological properties

If a topological space having some topological property implies its subspaces have that property, then we say the property is hereditary. If only closed subspaces must share the property we call it weakly hereditary.

  • Every open and every closed subspace of a completely metrizable space is completely metrizable.
  • Every open subspace of a Baire space is a Baire space.
  • Every closed subspace of a compact space is compact.
  • Being a Hausdorff space is hereditary.
  • Being a normal space is weakly hereditary.
  • Total boundedness is hereditary.
  • Being totally disconnected is hereditary.
  • First countability and second countability are hereditary.

Metric space aimed at its subspace

In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces.

Following (Holsztyński 1966), a notion of a metric space Y aimed at its subspace X is defined.

Informal introduction

Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X.

A priori, it may seem plausible that for a given X the superspaces Y that aim at X can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces which aim at a subspace isometric to X, there is a unique (up to isometry) universal one, Aim(X), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) X. And in the special case of an arbitrary compact metric space X every bounded subspace of an arbitrary metric space Y aimed at X is totally bounded (i.e. its metric completion is compact).

Definitions

Let be a metric space. Let be a subset of , so that (the set with the metric from restricted to ) is a metric subspace of . Then

Definition.  Space aims at if and only if, for all points of , and for every real , there exists a point of such that

Let be the space of all real valued metric maps (non-contractive) of . Define

Then

for every is a metric on . Furthermore, , where , is an isometric embedding of into ; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces into , where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space is aimed at .

Properties

Let be an isometric embedding. Then there exists a natural metric map such that :

for every and .

Theorem The space Y above is aimed at subspace X if and only if the natural mapping is an isometric embedding.

Thus it follows that every space aimed at X can be isometrically mapped into Aim(X), with some additional (essential) categorical requirements satisfied.

The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space M, which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of M onto Aim(X) (Holsztyński 1966).

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