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In mathematics, informally speaking, a 'limit point' of a set ''S'' in a topological space ''X'' is a point ''x'' in ''X'' that can be "approximated" by points of ''S'' other than ''x'' as well as one pleases. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by adding its limit points.
A related concept is 'cluster point' or 'accumulation point' of a sequence.
Let ''S'' be a subset of a topological space ''X''.
We say that a point ''x'' in ''X'' is a 'limit point' of ''S'' if
every open set containing ''x'' also contains a point of ''S'' other than ''x'' itself. This is equivalent to requiring that every neighbourhood of ''x'' contains a point of ''S'' other than ''x'' itself. (It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point.)
If every open set containing ''x'' contains infinitely many points of ''S'' then ''x'' is a specific type of limit point called a 'ω-accumulation point of ''S'''.
If every open set containing ''x'' contains uncountably many points of ''S'' then ''x'' is a specific type of limit point called a 'condensation point of ''S'''.
If ''X'' is a metric space, then a point ''x'' in ''X'' is a 'cluster point' of a sequence (''xn'' ) if for every ''ε'' > 0, there are infinitely many values of ''n'' such that ''d'' (''x'',''xn'' ) < ''ε''. Equivalently, that every neighborhood of ''x'' contains ''xn'' for infinitely many ''n''.
A limit point of the set of points in a sequence is a cluster point of the sequence. However, if for infinitely many ''n'' the values of ''xn'' are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence.
A cluster point of a sequence is a subsequential limit: the limit of some subsequence.
The concept of a net generalizes the idea of a sequence. Cluster points in nets encompass the idea of both condensation points and ω-accumulation points:
If φ is a net on ''X'' based on directed set ''D'' and ''A'' is a subset of ''X'', then φ is frequently in ''A'' if for every α in ''D'' there exists some β ≥ α, β in ''D'', so that φ(β) is in ''A''. A point ''x'' in ''X'' is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood ''U'' of ''x'', the net is frequently in ''U''.
Clustering and limit points are also defined for the related topic of filters.
The set of all cluster points of a sequence is sometimes called a limit set.
★ We have the following characterisation of limit points: ''x'' is a limit point of ''S'' if and only if it is in the closure of ''S'' {''x''}.
★
★ ''Proof'': We use the fact that a point is in the closure of a set if and only if every neighbourhood of the point contains a point of the set. Now, ''x'' is a limit point of ''S'', if and only if every neighbourhood of ''x'' contains a point of ''S'' other than ''x'', if and only if every neighbourhood of ''x'' contains a point of ''S'' {''x''}, if and only if ''x'' is in the closure of ''S'' {''x''}.
★ If we use L(''S'') to denote the set of limit points of ''S'', then we have the following characterisation of the closure of ''S'': The closure of ''S'' is equal to the union of ''S'' and L(''S'').
★
★ ''Proof'': ("Left subset") Suppose ''x'' is in the closure of ''S''. If ''x'' is in ''S'', we are done. If ''x'' is not in ''S'', then every neighbourhood of ''x'' contains a point of ''S'', and this point cannot be ''x''. In other words, ''x'' is a limit point of ''S'' and ''x'' is in L(''S''). ("Right subset") If ''x'' is in ''S'', then every neighbourhood of ''x'' clearly meets ''S'', so ''x'' is in the closure of ''S''. If ''x'' is in L(''S''), then every neighbourhood of ''x'' contains a point of ''S'' (other than ''x''), so ''x'' is again in the closure of ''S''. This completes the proof.
★ A corollary of this result gives us a characterisation of closed sets: A set ''S'' is closed if and only if it contains all of its limit points.
★
★ ''Proof'': ''S'' is closed if and only if ''S'' is equal to its closure if and only if ''S'' = ''S'' ∪ L(''S'') if and only if L(''S'') is contained in ''S''.
★
★ ''Another proof'': Let ''S'' be a closed set and ''x'' a limit point of ''S''. Then ''x'' must be in ''S'', for otherwise the complement of ''S'' would be an open neighborhood of ''x'' that does not intersect ''S''. Conversely, assume ''S'' contains all its limit points. We shall show that the complement of ''S'' is an open set. Let ''x'' be a point in the complement of ''S''. By assumption, ''x'' is not a limit point, and hence there exists an open neighborhood ''U'' of ''x'' that does not intersect ''S'', and so ''U'' lies entirely in the complement of ''S''. Hence the complement of ''S'' is open.
★ No isolated point is a limit point of any set.
★
★ ''Proof'': If ''x'' is an isolated point, then {''x''} is a neighbourhood of ''x'' that contains no points other than ''x''.
★ A space ''X'' is discrete if and only if no subset of ''X'' has a limit point.
★
★ ''Proof'': If ''X'' is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if ''X'' is not discrete, then there is a singleton {''x''} that is not open. Hence, every open neighbourhood of {''x''} contains a point ''y'' ≠ ''x'', and so ''x'' is a limit point of ''X''.
★ If a space ''X'' has the trivial topology and ''S'' is a subset of ''X'' with more than one element, then all elements of ''X'' are limit points of ''S''. If ''S'' is a singleton, then every point of ''X'' ''S'' is still a limit point of ''S''.
★
★ ''Proof'': As long as ''S'' {''x''} is nonempty, its closure will be ''X''. It's only empty when ''S'' is empty or ''x'' is the unique element of ''S''.
★ If a sequence in a space ''X'' converges to a point ''L'' in ''X'' and the elements are different from ''L'' then ''L'' is a limit point of ''X''.
★ If ''L'' is a limit point of space ''X'' then there is not necessarily a sequence in ''X''{L} converging to ''L''. For example, the smallest uncountable ordinal number (ω1) is a limit point (with respect to the order topology) of the set of countable ordinal numbers, but a convergent sequence of countable sets has a limit that is not larger than the union of those sets, which is countable.
★ However, if ''L'' is a limit point of ''metric'' space ''X'' then there ''is'' a sequence in ''X''{L} converging to ''L''.
★ Adherent point
★
A related concept is 'cluster point' or 'accumulation point' of a sequence.
| Contents |
| Definition |
| Types of limit points |
| Cluster point |
| Some facts |
| See also |
| References |
Definition
Let ''S'' be a subset of a topological space ''X''.
We say that a point ''x'' in ''X'' is a 'limit point' of ''S'' if
every open set containing ''x'' also contains a point of ''S'' other than ''x'' itself. This is equivalent to requiring that every neighbourhood of ''x'' contains a point of ''S'' other than ''x'' itself. (It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point.)
Types of limit points
If every open set containing ''x'' contains infinitely many points of ''S'' then ''x'' is a specific type of limit point called a 'ω-accumulation point of ''S'''.
If every open set containing ''x'' contains uncountably many points of ''S'' then ''x'' is a specific type of limit point called a 'condensation point of ''S'''.
Cluster point
If ''X'' is a metric space, then a point ''x'' in ''X'' is a 'cluster point' of a sequence (''xn'' ) if for every ''ε'' > 0, there are infinitely many values of ''n'' such that ''d'' (''x'',''xn'' ) < ''ε''. Equivalently, that every neighborhood of ''x'' contains ''xn'' for infinitely many ''n''.
A limit point of the set of points in a sequence is a cluster point of the sequence. However, if for infinitely many ''n'' the values of ''xn'' are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence.
A cluster point of a sequence is a subsequential limit: the limit of some subsequence.
The concept of a net generalizes the idea of a sequence. Cluster points in nets encompass the idea of both condensation points and ω-accumulation points:
If φ is a net on ''X'' based on directed set ''D'' and ''A'' is a subset of ''X'', then φ is frequently in ''A'' if for every α in ''D'' there exists some β ≥ α, β in ''D'', so that φ(β) is in ''A''. A point ''x'' in ''X'' is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood ''U'' of ''x'', the net is frequently in ''U''.
Clustering and limit points are also defined for the related topic of filters.
The set of all cluster points of a sequence is sometimes called a limit set.
Some facts
★ We have the following characterisation of limit points: ''x'' is a limit point of ''S'' if and only if it is in the closure of ''S'' {''x''}.
★
★ ''Proof'': We use the fact that a point is in the closure of a set if and only if every neighbourhood of the point contains a point of the set. Now, ''x'' is a limit point of ''S'', if and only if every neighbourhood of ''x'' contains a point of ''S'' other than ''x'', if and only if every neighbourhood of ''x'' contains a point of ''S'' {''x''}, if and only if ''x'' is in the closure of ''S'' {''x''}.
★ If we use L(''S'') to denote the set of limit points of ''S'', then we have the following characterisation of the closure of ''S'': The closure of ''S'' is equal to the union of ''S'' and L(''S'').
★
★ ''Proof'': ("Left subset") Suppose ''x'' is in the closure of ''S''. If ''x'' is in ''S'', we are done. If ''x'' is not in ''S'', then every neighbourhood of ''x'' contains a point of ''S'', and this point cannot be ''x''. In other words, ''x'' is a limit point of ''S'' and ''x'' is in L(''S''). ("Right subset") If ''x'' is in ''S'', then every neighbourhood of ''x'' clearly meets ''S'', so ''x'' is in the closure of ''S''. If ''x'' is in L(''S''), then every neighbourhood of ''x'' contains a point of ''S'' (other than ''x''), so ''x'' is again in the closure of ''S''. This completes the proof.
★ A corollary of this result gives us a characterisation of closed sets: A set ''S'' is closed if and only if it contains all of its limit points.
★
★ ''Proof'': ''S'' is closed if and only if ''S'' is equal to its closure if and only if ''S'' = ''S'' ∪ L(''S'') if and only if L(''S'') is contained in ''S''.
★
★ ''Another proof'': Let ''S'' be a closed set and ''x'' a limit point of ''S''. Then ''x'' must be in ''S'', for otherwise the complement of ''S'' would be an open neighborhood of ''x'' that does not intersect ''S''. Conversely, assume ''S'' contains all its limit points. We shall show that the complement of ''S'' is an open set. Let ''x'' be a point in the complement of ''S''. By assumption, ''x'' is not a limit point, and hence there exists an open neighborhood ''U'' of ''x'' that does not intersect ''S'', and so ''U'' lies entirely in the complement of ''S''. Hence the complement of ''S'' is open.
★ No isolated point is a limit point of any set.
★
★ ''Proof'': If ''x'' is an isolated point, then {''x''} is a neighbourhood of ''x'' that contains no points other than ''x''.
★ A space ''X'' is discrete if and only if no subset of ''X'' has a limit point.
★
★ ''Proof'': If ''X'' is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if ''X'' is not discrete, then there is a singleton {''x''} that is not open. Hence, every open neighbourhood of {''x''} contains a point ''y'' ≠ ''x'', and so ''x'' is a limit point of ''X''.
★ If a space ''X'' has the trivial topology and ''S'' is a subset of ''X'' with more than one element, then all elements of ''X'' are limit points of ''S''. If ''S'' is a singleton, then every point of ''X'' ''S'' is still a limit point of ''S''.
★
★ ''Proof'': As long as ''S'' {''x''} is nonempty, its closure will be ''X''. It's only empty when ''S'' is empty or ''x'' is the unique element of ''S''.
★ If a sequence in a space ''X'' converges to a point ''L'' in ''X'' and the elements are different from ''L'' then ''L'' is a limit point of ''X''.
★ If ''L'' is a limit point of space ''X'' then there is not necessarily a sequence in ''X''{L} converging to ''L''. For example, the smallest uncountable ordinal number (ω1) is a limit point (with respect to the order topology) of the set of countable ordinal numbers, but a convergent sequence of countable sets has a limit that is not larger than the union of those sets, which is countable.
★ However, if ''L'' is a limit point of ''metric'' space ''X'' then there ''is'' a sequence in ''X''{L} converging to ''L''.
See also
★ Adherent point
References
★
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