Locally operator - Revision history

Vipul: /* "locally" at a point */

2019-07-06T14:23:55Z

"locally" at a point

← Older revision		Revision as of 14:23, 6 July 2019
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	=== "locally" at a point ===		=== "locally" at a point ===

	Let <math>p</math> be a property of topological spaces. A topological space <math>X</math> is said to satisfy <math>p</math> ''locally'' at a point <math>x \in X</math> if the following holds: for any open subset <math>V \subseteq X</math> such that <math>x \in U</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U \subseteq V</math>, and <math>U</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>. The key difference with the global definition is that we are now imposing the condition relative to that particular point and not for all points.		Let <math>p</math> be a property of topological spaces. A topological space <math>X</math> is said to satisfy <math>p</math> ''locally'' at a point <math>x \in X</math> if the following holds: for any open subset <math>V \subseteq X</math> such that <math>x \in V</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U \subseteq V</math>, and <math>U</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>. The key difference with the global definition is that we are now imposing the condition relative to that particular point and not for all points.

	==Facts==		==Facts==

Vipul: /* Facts */

2019-07-06T14:17:45Z

Facts

← Older revision		Revision as of 14:17, 6 July 2019
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	* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Hausdorff space]], and locally <math>p</math> is the property of being a [[locally Hausdorff space]].		* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Hausdorff space]], and locally <math>p</math> is the property of being a [[locally Hausdorff space]].
	* If all spaces satisfying the property <math>p</math> are [[self-based space]]s, then being locally <math>p</math> is equivalent ~~tot he~~ condition that every point is contained an an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Euclidean space]], and locally <math>p</math> is the property of being a [[locally Euclidean space]].		* If all spaces satisfying the property <math>p</math> are [[self-based space]]s, then being locally <math>p</math> is equivalent to the condition that every point is contained an an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Euclidean space]], and locally <math>p</math> is the property of being a [[locally Euclidean space]].
	* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.		* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.
	* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.		* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.
	* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.		* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.

Vipul: /* Facts */

2016-05-30T20:57:27Z

Facts

← Older revision		Revision as of 20:57, 30 May 2016
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	==Facts==		==Facts==

	* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>.		* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Hausdorff space]], and locally <math>p</math> is the property of being a [[locally Hausdorff space]].
	* If all spaces satisfying the property <math>p</math> are [[self-based space]]s, then being locally <math>p</math> is equivalent tot he condition that every point is contained an an open subset satisfying <math>p</math>.		* If all spaces satisfying the property <math>p</math> are [[self-based space]]s, then being locally <math>p</math> is equivalent tot he condition that every point is contained an an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Euclidean space]], and locally <math>p</math> is the property of being a [[locally Euclidean space]].
	* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.		* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.
	* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.		* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.
	* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.		* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.

Vipul: /* Facts */

2016-05-30T20:55:59Z

Facts

← Older revision		Revision as of 20:55, 30 May 2016
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	* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>.		* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>.
			* If all spaces satisfying the property <math>p</math> are [[self-based space]]s, then being locally <math>p</math> is equivalent tot he condition that every point is contained an an open subset satisfying <math>p</math>.
	* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.		* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.
	* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.		* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.
	* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.		* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.

Vipul: /* "locally" at a point */

2016-05-30T20:52:12Z

"locally" at a point

← Older revision		Revision as of 20:52, 30 May 2016
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	=== "locally" at a point ===		=== "locally" at a point ===

	Let <math>p</math> be a property of topological spaces. A topological space <math>X</math> is said to satisfy <math>p</math> ''locally'' at a point <math>x \in X</math> if the following holds: for any open subset <math>U \subseteq X</math> such that <math>x \in U</math>, there exists an open subset <math>V</math> of <math>X</math> such that <math>x \in V \subseteq U</math>, and <math>V</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>.		Let <math>p</math> be a property of topological spaces. A topological space <math>X</math> is said to satisfy <math>p</math> ''locally'' at a point <math>x \in X</math> if the following holds: for any open subset <math>V \subseteq X</math> such that <math>x \in U</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U \subseteq V</math>, and <math>U</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>. The key difference with the global definition is that we are now imposing the condition relative to that particular point and not for all points.

	==Facts==		==Facts==

Vipul: /* Global "locally" operator */

2016-05-30T20:51:24Z

Global "locally" operator

← Older revision		Revision as of 20:51, 30 May 2016
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	# <math>X</math> has a [[basis for a topological space\|basis]] such that all the members of the basis satisfy property <math>p</math> when endowed with the [[subspace topology]].		# <math>X</math> has a [[basis for a topological space\|basis]] such that all the members of the basis satisfy property <math>p</math> when endowed with the [[subspace topology]].
	# For any point <math>x \in X</math> and any open subset <math>U \subseteq X</math> such that <math>x \in U</math>, there exists an open subset <math>V</math> of <math>X</math> such that <math>x \in V \subseteq U</math>, and <math>V</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>.		# For any point <math>x \in X</math> and any open subset <math>V \subseteq X</math> such that <math>x \in V</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U \subseteq X</math>, and <math>U</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>.

	=== "locally" at a point ===		=== "locally" at a point ===

Vipul: /* Definition */

2016-05-30T20:32:58Z

Definition

← Older revision		Revision as of 20:32, 30 May 2016
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	===Global "locally" operator ===		===Global "locally" operator ===

	Let <math>p</math> be a property of topological spaces. The property ''locally'' <math>p</math> is defined as follows~~: a~~ topological space <math>X</math> is termed locally <math>p</math> if it satisfies the following equivalent conditions:		Let <math>p</math> be a property of topological spaces. The property ''locally'' <math>p</math> is defined as follows. A topological space <math>X</math> is termed locally <math>p</math> if it satisfies the following equivalent conditions:

	# <math>X</math> has a [[basis for a topological space\|basis]] such that all the members of the basis satisfy property <math>p</math> when endowed with the [[subspace topology]].		# <math>X</math> has a [[basis for a topological space\|basis]] such that all the members of the basis satisfy property <math>p</math> when endowed with the [[subspace topology]].

Vipul: /* Facts */

2016-05-30T20:32:31Z

Facts

← Older revision		Revision as of 20:32, 30 May 2016
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	==Facts==		==Facts==

			* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>.
	* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.		* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.
	* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.		* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.
	* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.		* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.

Vipul at 20:29, 30 May 2016

2016-05-30T20:29:18Z

← Older revision		Revision as of 20:29, 30 May 2016
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	==Definition==		==Definition==

	Let <math>p</math> be a property of topological spaces. The property ''locally'' <math>p</math>, is defined as the ~~property of being~~ a topological space in ~~which every point has~~ an open ~~neighbourhood~~, ~~which, in~~ the [[subspace topology]], satisfies <math>p</math>.		===Global "locally" operator ===

			Let <math>p</math> be a property of topological spaces. The property ''locally'' <math>p</math> is defined as follows: a topological space <math>X</math> is termed locally <math>p</math> if it satisfies the following equivalent conditions:

			# <math>X</math> has a [[basis for a topological space\|basis]] such that all the members of the basis satisfy property <math>p</math> when endowed with the [[subspace topology]].
			# For any point <math>x \in X</math> and any open subset <math>U \subseteq X</math> such that <math>x \in U</math>, there exists an open subset <math>V</math> of <math>X</math> such that <math>x \in V \subseteq U</math>, and <math>V</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>.

			=== "locally" at a point ===

			Let <math>p</math> be a property of topological spaces. A topological space <math>X</math> is said to satisfy <math>p</math> ''locally'' at a point <math>x \in X</math> if the following holds: for any open subset <math>U \subseteq X</math> such that <math>x \in U</math>, there exists an open subset <math>V</math> of <math>X</math> such that <math>x \in V \subseteq U</math>, and <math>V</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>.

	==Facts==		==Facts==

			* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <math>p</math> implies locally <math>p</math>.
	* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.		* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.
	* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.		* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.

Vipul: /* Facts */

2016-05-30T20:18:47Z

Facts

← Older revision		Revision as of 20:18, 30 May 2016
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	==Facts==		==Facts==

	* For any property <math>p</math> of topological spaces, <math>p</math> implies locally <math>p</math>.
	* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.		* There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.
	* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.		* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.