Connected not implies path-connected - Revision history

Vipul: /* Partial truth */

2009-12-25T07:02:14Z

Partial truth

← Older revision		Revision as of 07:02, 25 December 2009
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	* For subsets of the [[real line]] (with the usual topology), the notions of connected and path-connected coincide. Specifically, the connected sets, which are also the path-connected sets, are precisely the intervals (open, closed, half-open half-closed, and possibly extending to infinity in one or both directions). Note that from our examples, it is clear that this breaks down for subsets of <math>\R^2</math>.		* For subsets of the [[real line]] (with the usual topology), the notions of connected and path-connected coincide. Specifically, the connected sets, which are also the path-connected sets, are precisely the intervals (open, closed, half-open half-closed, and possibly extending to infinity in one or both directions). Note that from our examples, it is clear that this breaks down for subsets of <math>\R^2</math>.
	* For [[locally connected space]]s, the notions of connectedness and path-connectedness coincide. This includes open subsets of Euclidean space, [[locally Euclidean space]]s, and in particular, [[manifold]]s.		* For [[locally path-connected space]]s, the notions of connectedness and path-connectedness coincide. This includes open subsets of Euclidean space, [[locally Euclidean space]]s, and in particular, [[manifold]]s.

			==Converse==

			The converse is true, i.e., [[path-connected implies connected]].

	==Proof==		==Proof==

Vipul at 06:58, 25 December 2009

2009-12-25T06:58:47Z

← Older revision		Revision as of 06:58, 25 December 2009
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	A [[topological space]] is termed '''path-connected''' if, given any two distinct points in the topological space, there is a path from one point to the other. Here, a [[path]] is a continuous function from the [[unit interval]] to the space, with the image of <math>0</math> being the ''starting point'' or ''source'' and the image of <math>1</math> being the ''ending point'' or ''terminus''.		A [[topological space]] is termed '''path-connected''' if, given any two distinct points in the topological space, there is a path from one point to the other. Here, a [[path]] is a continuous function from the [[unit interval]] to the space, with the image of <math>0</math> being the ''starting point'' or ''source'' and the image of <math>1</math> being the ''ending point'' or ''terminus''.

			==Partial truth==

			The following are true:

			* For subsets of the [[real line]] (with the usual topology), the notions of connected and path-connected coincide. Specifically, the connected sets, which are also the path-connected sets, are precisely the intervals (open, closed, half-open half-closed, and possibly extending to infinity in one or both directions). Note that from our examples, it is clear that this breaks down for subsets of <math>\R^2</math>.
			* For [[locally connected space]]s, the notions of connectedness and path-connectedness coincide. This includes open subsets of Euclidean space, [[locally Euclidean space]]s, and in particular, [[manifold]]s.

	==Proof==		==Proof==

Vipul: Created page with '{{topospace property non-implication| stronger = connected space| weaker = path-connected space}} ==Statement== It is possible for a topological space to be a [[connected s…'

2009-12-25T06:40:24Z

Created page with '{{topospace property non-implication| stronger = connected space| weaker = path-connected space}} ==Statement== It is possible for a topological space to be a [[connected s…'

New page

{{topospace property non-implication|
stronger = connected space|
weaker = path-connected space}}

==Statement==

It is possible for a [[topological space]] to be a [[connected space]] but ''not'' a [[path-connected space]].

==Definitions used==

===Connected space===

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

A [[topological space]] is termed '''connected''' if it cannot be expressed as a disjoint union of two nonempty open subsets.

===Path-connected space===

{{further|[[path-connected space]]}}

A [[topological space]] is termed '''path-connected''' if, given any two distinct points in the topological space, there is a path from one point to the other. Here, a [[path]] is a continuous function from the [[unit interval]] to the space, with the image of <math>0</math> being the ''starting point'' or ''source'' and the image of <math>1</math> being the ''ending point'' or ''terminus''.

==Proof==

===Examples of the topologist's sine curve===

{{further|[[particular example::topologist's sine curve]], [[particular example::closed topologist's sine curve]]}}

The topologist's sine curve is the union of the graph of the function <math>\sin(1/x)</math> for <math>x</math> in the interval <math>(0,1]</math> and the origin. (There are other variants -- for instance, the right endpoint is not always taken as <math>1</math> but can be any positive number). It acquires the [[subspace topology]] from the Euclidean plane.

* The topologist's sine curve is not path-connected: There is no path connecting the origin to any other point on the space.
* The topologist's sine curve is connected: All nonzero points are in the same connected component, so the only way it could be disconnected is if the origin and the rest of the space were the two connected components. But in that case, both the origin and the rest of the space would be open subsets, and the origin is not open in the space becaues an arbitrarily small ball around the origin intersects the rest of the space.

The closed topologist's sine curve is the closure, in the Euclidean plane, of the topologist's sine curve. It includes all points on the <math>y</math>-axis with <math>y</math>-coordinate between <math>-1</math> and <math>1</math>. For reasons similar to the above, the closed topologist's sine curve is connected but not path-connected.

===Example of the infinite broom===

{{further|[[particular example::infinite broom]]}}

The infinite broom is another example of a topological space that is connected but not path-connected.

Note that unlike the case of the topologist's sine curve, the closure of the infinite broom in the Euclidean plane, known as the [[closed infinite broom]] (also sometimes as the broom space) ''is'' a [[path-connected space]].