Khanh at 20:30, 23 January 2022

2022-01-23T20:30:49Z

Khanh: Created page with "{{Redirect|Induced topology|the topology generated by a family of functions|Initial topology}} In topology and related areas of mathematics, a '''subspace''' of a t..."

2022-01-23T19:23:16Z

Created page with "{{Redirect|Induced topology|the topology generated by a family of functions|Initial topology}} In topology and related areas of mathematics, a '''subspace''' of a t..."

New page

{{Redirect|Induced topology|the topology generated by a family of functions|Initial 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 [[Topological_space#Definitions|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 <math>(X, \tau)</math> and a [[subset]] <math>S</math> of <math>X</math>, the '''subspace topology''' on <math>S</math> is defined by
:<math>\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.</math>
That is, a subset of <math>S</math> is open in the subspace topology [[if and only if]] it is the [[intersection (set theory)|intersection]] of <math>S</math> with an [[open set]] in <math>(X, \tau)</math>. If <math>S</math> is equipped with the subspace topology then it is a topological space in its own right, and is called a '''subspace''' of <math>(X, \tau)</math>. 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 <math>S</math> of <math>X</math> as the [[coarsest topology]] for which the [[inclusion map]]
:<math>\iota: S \hookrightarrow X</math>
is [[continuous (topology)|continuous]].

More generally, suppose <math>\iota</math> is an [[Injective function|injection]] from a set <math>S</math> to a topological space <math>X</math>. Then the subspace topology on <math>S</math> is defined as the coarsest topology for which <math>\iota</math> is continuous. The open sets in this topology are precisely the ones of the form <math>\iota^{-1}(U)</math> for <math>U</math> open in <math>X</math>. <math>S</math> is then [[homeomorphic]] to its image in <math>X</math> (also with the subspace topology) and <math>\iota</math> is called a [[topological embedding]].

A subspace <math>S</math> is called an '''open subspace''' if the injection <math>\iota</math> is an [[open map]], i.e., if the forward image of an open set of <math>S</math> is open in <math>X</math>. Likewise it is called a '''closed subspace''' if the injection <math>\iota</math> 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 <math>S</math> is a subset of <math>X</math>, and <math>(X, \tau)</math> is a topological space, then the unadorned symbols "<math>S</math>" and "<math>X</math>" can often be used to refer both to <math>S</math> and <math>X</math> considered as two subsets of <math>X</math>, and also to <math>(S,\tau_S)</math> and <math>(X,\tau)</math> as the topological spaces, related as discussed above. So phrases such as "<math>S</math> an open subspace of <math>X</math>" are used to mean that <math>(S,\tau_S)</math> is an open subspace of <math>(X,\tau)</math>, in the sense used below; that is: (i) <math>S \in \tau</math>; and (ii) <math>S</math> is considered to be endowed with the subspace topology.

== Examples ==
In the following, <math>\mathbb{R}</math> represents the [[real number]]s with their usual topology.
* The subspace topology of the [[natural number]]s, as a subspace of <math>\mathbb{R}</math>, is the [[discrete topology]].
* The [[rational number]]s <math>\mathbb{Q}</math> considered as a subspace of <math>\mathbb{R}</math> do not have the discrete topology ({0} for example is not an open set in <math>\mathbb{Q}</math>). 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 <math>\mathbb{R}</math> is both open and closed, whereas as a subset of <math>\mathbb{R}</math> it is only closed.
* As a subspace of <math>\mathbb{R}</math>, [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 <math>\mathbb{R}</math>. Then [0, {{frac|1|2}}) is open in ''S'' but not in <math>\mathbb{R}</math>. Likewise [{{frac|1|2}}, 1) is closed in ''S'' but not in <math>\mathbb{R}</math>. ''S'' is both open and closed as a subset of itself but not as a subset of <math>\mathbb{R}</math>.

== Properties ==

The subspace topology has the following characteristic property. Let <math>Y</math> be a subspace of <math>X</math> and let <math>i : Y \to X</math> be the inclusion map. Then for any topological space <math>Z</math> a map <math>f : Z\to Y</math> is continuous [[if and only if]] the composite map <math>i\circ f</math> is continuous.
[[Image:Subspace-01.png|center|Characteristic property of the subspace topology]]
This property is characteristic in the sense that it can be used to define the subspace topology on <math>Y</math>.

We list some further properties of the subspace topology. In the following let <math>S</math> be a subspace of <math>X</math>.

* If <math>f:X\to Y</math> is continuous then the restriction to <math>S</math> is continuous.
* If <math>f:X\to Y</math> is continuous then <math>f:X\to f(X)</math> is continuous.
* The closed sets in <math>S</math> are precisely the intersections of <math>S</math> with closed sets in <math>X</math>.
* If <math>A</math> is a subspace of <math>S</math> then <math>A</math> is also a subspace of <math>X</math> with the same topology. In other words the subspace topology that <math>A</math> inherits from <math>S</math> is the same as the one it inherits from <math>X</math>.
* Suppose <math>S</math> is an open subspace of <math>X</math> (so <math>S\in\tau</math>). Then a subset of <math>S</math> is open in <math>S</math> if and only if it is open in <math>X</math>.
* Suppose <math>S</math> is a closed subspace of <math>X</math> (so <math>X\setminus S\in\tau</math>). Then a subset of <math>S</math> is closed in <math>S</math> if and only if it is closed in <math>X</math>.
* If <math>B</math> is a [[basis (topology)|basis]] for <math>X</math> then <math>B_S = \{U\cap S : U \in B\}</math> is a basis for <math>S</math>.
* The topology induced on a subset of a [[metric space]] by restricting the [[metric (mathematics)|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.

{{Short description|Universal property of metric spaces}}In [[mathematics]], a '''metric space aimed at its subspace''' is a [[category theory|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 space]]s.

Following {{harv|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 property|universal]] one, Aim(''X''), which in a sense of canonical [[isometric embedding]]s 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 <math>(Y, d)</math> be a metric space. Let <math>X</math> be a subset of <math>Y</math>, so that <math>(X,d |_X)</math> (the set <math>X</math> with the metric from <math>Y</math> restricted to <math>X</math>) is a metric subspace of <math>(Y,d)</math>. Then

'''Definition'''.  Space <math>Y</math> aims at <math>X</math> if and only if, for all points <math>y, z</math> of <math>Y</math>, and for every real <math>\epsilon > 0</math>, there exists a point <math>p</math> of <math>X</math> such that

:<math>|d(p,y) - d(p,z)| > d(y,z) - \epsilon.</math>

Let <math>\text{Met}(X)</math> be the space of all real valued [[metric map]]s (non-[[Contraction mapping|contractive]]) of <math>X</math>. Define

:<math>\text{Aim}(X) := \{f \in \operatorname{Met}(X) : f(p) + f(q) \ge d(p,q) \text{ for all } p,q\in X\}.</math>

Then

:<math>d(f,g) := \sup_{x\in X} |f(x)-g(x)| < \infty</math>

for every <math>f, g\in \text{Aim}(X)</math> is a metric on <math>\text{Aim}(X)</math>. Furthermore, <math>\delta_X\colon x\mapsto d_x</math>, where <math>d_x(p) := d(x,p)\,</math>, is an isometric embedding of <math>X</math> into <math>\operatorname{Aim}(X)</math>; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces <math>X</math> into <math>C(X)</math>, where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space <math>\operatorname{Aim}(X)</math> is aimed at <math>\delta_X(X)</math>.

== Properties ==
Let <math>i\colon X \to Y</math> be an isometric embedding. Then there exists a natural metric map <math>j\colon Y \to \operatorname{Aim}(X)</math> such that <math>j \circ i = \delta_X</math>:

:::<math>(j(y))(x) := d(x,y)\,</math>

for every <math>x\in X\,</math> and <math>y\in Y\,</math>.

:'''Theorem''' The space ''Y'' above is aimed at subspace ''X'' if and only if the natural mapping <math>j\colon Y \to \operatorname{Aim}(X)</math> 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 metric space|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) {{harv|Holsztyński|1966}}.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Subspace_topology Subspace topology, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Metric_space_aimed_at_its_subspace Metric space aimed at its subspace, Wikipedia] under a CC BY-SA license

Subspaces of Metric Spaces - Revision history

Khanh at 20:30, 23 January 2022

Khanh: Created page with "{{Redirect|Induced topology|the topology generated by a family of functions|Initial topology}} In topology and related areas of mathematics, a '''subspace''' of a t..."