Space of path components - Revision history

Vipul: /* As a monoid */

2019-07-06T13:47:25Z

As a monoid

← Older revision		Revision as of 13:47, 6 July 2019
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	===As a monoid===		===As a monoid===

	If <math>X</math> is a [[topological group]], [[topological monoid]], or (most generally) a [[H-space]], then <math>\pi_0(X)</math> naturally acquires the structure of a [[monoid]] ~~(in fact, a [[topological monoid]], but a discrete one if <math>X</math> is locally path-connected)~~. The structure is obtained by quotienting the original monoid (or H-space multiplication) by the equivalence relation of being path-connected.		If <math>X</math> is a [[topological group]], [[topological monoid]], or (most generally) a [[H-space]], then <math>\pi_0(X)</math> naturally acquires the structure of a [[monoid]]. The structure is obtained by quotienting the original monoid (or H-space multiplication) by the equivalence relation of being path-connected.

	In order to perform this quotienting, we need to justify that the relation of being in the same path component is a congruence for the original monoid (or H-space multiplication). This is indeed guaranteed by the continuity of multiplication: if <math>x_1</math> and <math>y_1</math> are in the same path component, and <math>x_2</math> and <math>y_2</math> are in the same path component, then <math>x_1 * x_2</math> and <math>y_1 * y_2</math> are in the same path component, because the paths can be multiplied pointwise.		In order to perform this quotienting, we need to justify that the relation of being in the same path component is a congruence for the original monoid (or H-space multiplication). This is indeed guaranteed by the continuity of multiplication: if <math>x_1</math> and <math>y_1</math> are in the same path component, and <math>x_2</math> and <math>y_2</math> are in the same path component, then <math>x_1 * x_2</math> and <math>y_1 * y_2</math> are in the same path component, because the paths can be multiplied pointwise.

	Further, even if the original multiplication was not associative but only associative up to homotopy (making it an H-space), the new multiplication is strictly associative because any homotopy must descend to the identity map at the level of path components.		Further, even if the original multiplication was not associative but only associative up to homotopy (making it an H-space), the new multiplication is strictly associative because any homotopy must descend to the identity map at the level of path components.

			'''However, the multiplication need not be continuous on <math>\pi_0(X)</math>.''' Therefore, <math>\pi_0(X)</math> nee not acquire the structure of a topological monoid.

			== References ==

			Some work of J. Brazas or P. Fabel may be relevant.

Vipul at 04:36, 31 December 2010

2010-12-31T04:36:12Z

← Older revision		Revision as of 04:36, 31 December 2010
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	==Definition==		==Definition==

	===As a set and ~~topological space~~===		===As a set and pointed set===

	Suppose <math>X</math> is a [[topological space]] (we sometimes take a [[based topological space]], but the basepoint turns out to be irrelevant). The '''~~space~~ of path components''' of <math>X</math>, denoted <math>\pi_0(X)</math>, is defined as follows:		Suppose <math>X</math> is a [[topological space]] (we sometimes take a [[based topological space]], but the basepoint turns out to be irrelevant). The '''set of path components''' of <math>X</math>, denoted <math>\pi_0(X)</math>, is defined as follows:

	* As a set, it is the set of [[path component]]s of <math>X</math>. In other words, it is the set of equivalence classes of <math>X</math> under the equivalence relation <math>x \sim y</math> iff there is a [[path]] from <math>x</math> to <math>y</math> in <math>X</math>.		* As a set, it is the set of [[path component]]s of <math>X</math>. In other words, it is the set of equivalence classes of <math>X</math> under the equivalence relation <math>x \sim y</math> iff there is a [[path]] from <math>x</math> to <math>y</math> in <math>X</math>.
	* As a [[topological space]], it is obtained by taking the [[quotient topology]] of <math>X</math> under the equivalence relation <math>x \sim y</math> iff there is a path from <math>x</math> to <math>y</math> in <math>X</math>.		* If <math>X</math> is nonempty and <math>x \in X</math>, then <math>\pi_0(X,x)</math>, as a ''pointed set'', is defined as the set <math>\pi_0(X)</math> with the chosen basepoint being the element of <math>\pi_0(X)</math> that is the path component of <math>x</math>.

			===As a topological space===

			As a [[topological space]], the '''space of path components''' is obtained by taking the [[quotient topology]] of <math>X</math> under the equivalence relation <math>x \sim y</math> iff there is a path from <math>x</math> to <math>y</math> in <math>X</math>.

	For a [[locally path-connected space]], the space of path components is a [[discrete space]]. Thus, often, when dealing with such spaces, we ignore the topology.		For a [[locally path-connected space]], the space of path components is a [[discrete space]]. Thus, often, when dealing with such spaces, we ignore the topology.

Vipul: Created page with '==Definition== ===As a set and topological space=== Suppose $X$ is a topological space (we sometimes take a based topological space, but the basepoint turns ...'

2010-12-20T00:05:40Z

Created page with '==Definition== ===As a set and topological space=== Suppose <math>X</math> is a topological space (we sometimes take a based topological space, but the basepoint turns ...'

New page

==Definition==

===As a set and topological space===

Suppose <math>X</math> is a [[topological space]] (we sometimes take a [[based topological space]], but the basepoint turns out to be irrelevant). The '''space of path components''' of <math>X</math>, denoted <math>\pi_0(X)</math>, is defined as follows:

* As a set, it is the set of [[path component]]s of <math>X</math>. In other words, it is the set of equivalence classes of <math>X</math> under the equivalence relation <math>x \sim y</math> iff there is a [[path]] from <math>x</math> to <math>y</math> in <math>X</math>.
* As a [[topological space]], it is obtained by taking the [[quotient topology]] of <math>X</math> under the equivalence relation <math>x \sim y</math> iff there is a path from <math>x</math> to <math>y</math> in <math>X</math>.

For a [[locally path-connected space]], the space of path components is a [[discrete space]]. Thus, often, when dealing with such spaces, we ignore the topology.

===As a monoid===

If <math>X</math> is a [[topological group]], [[topological monoid]], or (most generally) a [[H-space]], then <math>\pi_0(X)</math> naturally acquires the structure of a [[monoid]] (in fact, a [[topological monoid]], but a discrete one if <math>X</math> is locally path-connected). The structure is obtained by quotienting the original monoid (or H-space multiplication) by the equivalence relation of being path-connected.

In order to perform this quotienting, we need to justify that the relation of being in the same path component is a congruence for the original monoid (or H-space multiplication). This is indeed guaranteed by the continuity of multiplication: if <math>x_1</math> and <math>y_1</math> are in the same path component, and <math>x_2</math> and <math>y_2</math> are in the same path component, then <math>x_1 * x_2</math> and <math>y_1 * y_2</math> are in the same path component, because the paths can be multiplied pointwise.

Further, even if the original multiplication was not associative but only associative up to homotopy (making it an H-space), the new multiplication is strictly associative because any homotopy must descend to the identity map at the level of path components.

Space of path components - Revision history

Vipul: /* As a monoid */

Vipul at 04:36, 31 December 2010

Vipul: Created page with '==Definition== ===As a set and topological space=== Suppose X is a topological space (we sometimes take a based topological space, but the basepoint turns ...'

Vipul: Created page with '==Definition== ===As a set and topological space=== Suppose $X$ is a topological space (we sometimes take a based topological space, but the basepoint turns ...'