Connectedness is product-closed - Revision history

Vipul: /* Proof details */

2015-11-16T01:16:34Z

Proof details

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	'''ASSUMPTION''': Suppose <math>X</math> is a disjoint union of two nonempty open subsets <math>U</math> and <math>V</math>.		'''ASSUMPTION''': Suppose <math>X</math> is a disjoint union of two nonempty open subsets <math>U</math> and <math>V</math>.

	We then try to derive a contradiction. Note that Steps (1)-(3) do not make use of the assumption, and are independently true.		We then try to derive a contradiction. Note that Steps (1)-(3) do not make use of the assumption, and are independently true. Also, note that the definition of product topology really gets used in Step (5). Everything until that point is also true for the box topology.

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Proof outline */

2015-11-16T01:15:57Z

Proof outline

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	Now any partition of the whole space into disjoint open subsets must partition each slice into disjoint open subsets; but since each slice is connected, each slice must lie in one of the parts. This allows us to show that if two points differ in only finitely many coordinates, then they must lie in the same open subset of the partition.		Now any partition of the whole space into disjoint open subsets must partition each slice into disjoint open subsets; but since each slice is connected, each slice must lie in one of the parts. This allows us to show that if two points differ in only finitely many coordinates, then they must lie in the same open subset of the partition.

	Finally, we use the fact that any open set must contain a basis open set; the basis open set allows us to alter the remaining ''cofinitely'' many coordinates.		Finally, we use the fact that any open set must contain a basis open set; the basis open set allows us to alter the remaining ''cofinitely'' many coordinates. Note that it is this part that crucially uses the definition of product topology, and it is the analogous step to this that would fail for the box topology.

	===Proof details===		===Proof details===

Vipul: /* Proof details */

2015-11-16T01:12:47Z

Proof details

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	===Proof details===		===Proof details===

	'''Given''': An indexing set <math>I</math>, a family <math>(X_i)_{i \in I}</math> of topological spaces. <math>X</math> is the product of these, with the product topology		'''Given''': An indexing set <math>I</math>, a family <math>(X_i)_{i \in I}</math> of nonempty topological spaces. <math>X</math> is the product of these, with the product topology

	'''To prove''': <math>X</math> is a connected space. In other words, <math>X</math> cannot be expressed as a disjoint union of two nonempty open subsets <math>U</math> and <math>V</math>		'''To prove''': <math>X</math> is a connected space. In other words, <math>X</math> cannot be expressed as a disjoint union of two nonempty open subsets <math>U</math> and <math>V</math>

Vipul: /* Statement */

2015-11-16T01:12:39Z

Statement

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	==Statement==		==Statement==

	Suppose <math>X_i, i \in I</math> is a family of [[topological space]]s, all of which are [[connected space]]s. Then, the product space <math>\prod_{i \in I} X_i</math>, endowed with the [[product topology]], is also a connected space.		Suppose <math>X_i, i \in I</math> is a family of nonempty [[topological space]]s, all of which are [[connected space]]s. Then, the product space <math>\prod_{i \in I} X_i</math>, endowed with the [[product topology]], is also a connected space.

	==Related facts==		==Related facts==

Vipul at 01:12, 16 November 2015

2015-11-16T01:12:19Z

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	==Statement==		==Statement==

	~~===Property-theoretic statement===~~		Suppose <math>X_i, i \in I</math> is a family of [[topological space]]s, all of which are [[connected space]]s. Then, the product space <math>\prod_{i \in I} X_i</math>, endowed with the [[product topology]], is also a connected space.

	~~The [[property of topological spaces]] of being [[connected space\|connected]] is a [[product-closed property of topological spaces]].~~		==Related facts==

	~~===Verbal statement===~~		* [[Compactness is product-closed]]
			* [[Hausdorffness is product-closed]]
	~~An arbitrary product of~~ [[~~connected space]]s~~ is ~~connected, in the [[~~product ~~topology]].~~

	~~==Definitions used==~~

	~~===Connected space===~~

	~~{{further\|[[connected space~~]]}}

	~~===Product topology===~~
	~~{{further\|~~[[product ~~topology~~]]}}

	==Proof==		==Proof==

Vipul: /* Proof details */

2012-01-26T17:40:27Z

Proof details

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	'''Given''': An indexing set <math>I</math>, a family <math>(X_i)_{i \in I}</math> of topological spaces. <math>X</math> is the product of these, with the product topology		'''Given''': An indexing set <math>I</math>, a family <math>(X_i)_{i \in I}</math> of topological spaces. <math>X</math> is the product of these, with the product topology

	'''To prove''': <math>X</math> is a connected space		'''To prove''': <math>X</math> is a connected space. In other words, <math>X</math> cannot be expressed as a disjoint union of two nonempty open subsets <math>U</math> and <math>V</math>

	'''Proof''': ~~First, observe that for fixed values of other coordinates,~~ the ~~inclusion~~ from ~~<math>X_i</math> to <math>X</math> is a subspace embedding, so its image is connected. In other words, every ''slice'' is connected~~. ~~Further, using slices along different <math>X_i</math>, we see that if two points of <math>X</math> differ in only finitely many coordinates, they are in~~ the ~~same connected component.~~		'''Proof''': We use a proof by contradiction (the contradiction starts building from Step (4)). Explicitly westart with the assumption:

	~~Now, suppose~~ <math>X</math> is a disjoint union of two nonempty open subsets <math>U</math> and ~~<math>V</math>. Then, by the above, it is clear that if two points of <math>X</math> differ in only finitely many coordinates, they must either both lie in <math>U</math> or both lie in~~ <math>V</math>.		'''ASSUMPTION''': Suppose <math>X</math> is a disjoint union of two nonempty open subsets <math>U</math> and <math>V</math>.

	~~Now~~, ~~since~~ <math>U</math> is open, it contains a basis element -- say, an open ~~subet~~ <math>W</math> given by <math>\~~times~~ W_i</math> where finitely many of the <math>W_i</math> are not the whole space. Let's call these finite ones <math>i_1, i_2, \dots, i_n</math>. For any point of <math>X</math>, we can thus find a point of <math>W</math> that differs from it in at most ~~these~~ <math>n</math> coordinates. By the ~~above~~, ~~any~~ point of <math>X</math> ~~must be~~ in <math>U</math>, ~~contradicting~~ the assumption that <math>U</math> ~~and~~ <math>V</math> ~~are nonempty open subsets whose disjoint union is~~ <math>X</math>.		We then try to derive a contradiction. Note that Steps (1)-(3) do not make use of the assumption, and are independently true.

			{\| class="sortable" border="1"
			! Step no. !! Assertion/construction !! Facts used!! Given data used !! Previous steps used !! Explanation
			\|-
			\| 1 \|\| Choose <math>i \in I</math> and fix the values of all coordinates except the <math>i^{th}</math> coordinate. Consider the inclusion from <math>X_i</math> to <math>X</math> that sends a point of <math>X_i</math> to the tuple where all other coordinates are the pre-determined ones. This is a subspace embedding and in particular, the image of <math>X_i</math> is a connected subset in <math>X</math>. \|\| \|\| \|\| \|\|
			\|-
			\| 2 \|\| If two elements of <math>X</math> differ in only one coordinate, they are in the same connected component. \|\| \|\| \|\| Step (1) \|\| Step-direct -- the image of the corresponding <math>X_i</math> is a connected subset containing both.
			\|-
			\| 3 \|\| If two elements of <math>X</math> differ in only finitely many coordinates, they are in the same connected component. \|\| \|\| \|\| Step (2) \|\| If two elements differ in finitely many coordinates, we can construct a sequence of elements going from one to the other where adjacent members of the sequence differ in only one coordinate. Any two adjacent members of the sequence are in the same connected component, and thus, all members of the sequence are in the same connected component.
			\|-
			\| 4 \|\| If two points of <math>X</math> differ in only finitely many coordinates, they must either both lie in <math>U</math> or both lie in <math>V</math>. \|\| \|\| Assumption \|\| Step (3) \|\| By Step (3), any two points differing in only finitely many coordinates lie in the same connected component, say <math>C</math>. However, if one lies in <math>U</math> and the other lies in <math>V</math>, then <math>C</math> itself is a union of non-empty disjoint open subsets of itself <math>C \cap U</math> and <math>C \cap V</math>, making it disconnected. Thus, they must both lie in the same piece.
			\|-
			\| 5 \|\| Since <math>U</math> is open and non-empty, it contains a basis element -- say, an open subset <math>W</math> given by <math>\prod_{i \in I} W_i</math> where all the <math>W_i</math> are non-empty and only finitely many of the <math>W_i</math> are not the whole space. Let's call these finite ones <math>i_1, i_2, \dots, i_n</math>. \|\|definition of product topology \|\| \|\| \|\|
			\|-
			\| 6 \|\| For any point of <math>X</math>, we can thus find a point of <math>W</math> (and hence of <math>U</math>) that differs from it in at most the <math>n</math> coordinates found in Step (5). \|\| \|\| \|\| Step (5) \|\| Simply replace the <math>n</math> coordinates by coordinates of elements of <math>W_{i_1}, W_{i_2}, \dots, W_{i_n}</math> and leave the remaining coordinates as they are.
			\|-
			\| 7 \|\| Any point of <math>X</math> is in <math>U</math>, completing the contradiction to the assumption that <math>V</math> is nonempty. \|\| \|\| \|\| Steps (4), (6) \|\| By Step (6), we can, for any point in <math>X</math>, find an element of <math>U</math> that differs from it in finitely many coordinates. Thus, by Step (4), the point in <math>X</math> that we start with must also be in <math>U</math>.
			\|}

	==References==		==References==

Vipul at 11:21, 8 August 2008

2008-08-08T11:21:28Z

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	{{topospace metaproperty satisfaction}}		{{topospace metaproperty satisfaction\|
			property = connected space\|
			metaproperty = product-closed property of topological spaces}}

	==Statement==		==Statement==

Vipul: /* Proof */

2008-07-20T19:43:56Z

Proof

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	Finally, we use the fact that any open set must contain a basis open set; the basis open set allows us to alter the remaining ''cofinitely'' many coordinates.		Finally, we use the fact that any open set must contain a basis open set; the basis open set allows us to alter the remaining ''cofinitely'' many coordinates.

			===Proof details===

			'''Given''': An indexing set <math>I</math>, a family <math>(X_i)_{i \in I}</math> of topological spaces. <math>X</math> is the product of these, with the product topology

			'''To prove''': <math>X</math> is a connected space

			'''Proof''': First, observe that for fixed values of other coordinates, the inclusion from <math>X_i</math> to <math>X</math> is a subspace embedding, so its image is connected. In other words, every ''slice'' is connected. Further, using slices along different <math>X_i</math>, we see that if two points of <math>X</math> differ in only finitely many coordinates, they are in the same connected component.

			Now, suppose <math>X</math> is a disjoint union of two nonempty open subsets <math>U</math> and <math>V</math>. Then, by the above, it is clear that if two points of <math>X</math> differ in only finitely many coordinates, they must either both lie in <math>U</math> or both lie in <math>V</math>.

			Now, since <math>U</math> is open, it contains a basis element -- say, an open subet <math>W</math> given by <math>\times W_i</math> where finitely many of the <math>W_i</math> are not the whole space. Let's call these finite ones <math>i_1, i_2, \dots, i_n</math>. For any point of <math>X</math>, we can thus find a point of <math>W</math> that differs from it in at most these <math>n</math> coordinates. By the above, any point of <math>X</math> must be in <math>U</math>, contradicting the assumption that <math>U</math> and <math>V</math> are nonempty open subsets whose disjoint union is <math>X</math>.

	==References==		==References==

Vipul at 19:31, 20 July 2008

2008-07-20T19:31:49Z

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	Finally, we use the fact that any open set must contain a basis open set; the basis open set allows us to alter the remaining ''cofinitely'' many coordinates.		Finally, we use the fact that any open set must contain a basis open set; the basis open set allows us to alter the remaining ''cofinitely'' many coordinates.

			==References==

			===Textbook references===
			* {{booklink-proved\|Munkres}}, Page 150, Theorem 23.6 (does the case of finite products), adn Page 152, Exercise 10 (outlines a proof for infinite products), Chapter 3, Section 23

Vipul at 19:23, 20 July 2008

2008-07-20T19:23:24Z

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			{{topospace metaproperty satisfaction}}

	==Statement==		==Statement==