2.5 Connected spaces

Connected spaces are fundamental in topology, capturing the idea of a space being in one piece. They can't be divided into two separate open sets, making them crucial for understanding global structures in noncommutative geometry.

Path connectedness is a stronger property, allowing any two points to be joined by a continuous path. This concept helps analyze the structure of spaces and their relationships, playing a key role in studying noncommutative spaces.

Connected spaces

• Connected spaces are topological spaces that cannot be divided into two disjoint non-empty open sets
• Connectedness is a fundamental property in topology that captures the idea of a space being in one piece
• The study of connected spaces is crucial in noncommutative geometry as it helps understand the global structure and properties of spaces

Path connectedness

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• A space $X$ is path connected if for any two points $x, y \in X$, there exists a continuous function $f: [0, 1] \rightarrow X$ such that $f(0) = x$ and $f(1) = y$
• Path connectedness implies that any two points in the space can be joined by a continuous path
• Path connectedness is a stronger property than connectedness
  • Every path connected space is connected, but not every connected space is path connected (sine curve with a point)

Connectedness vs path connectedness

• Connectedness is a weaker property than path connectedness
• A space can be connected but not path connected (topologist's sine curve)
• However, every path connected space is necessarily connected
• In some special cases, such as locally path connected spaces, connectedness and path connectedness are equivalent

Connected components

• A connected component of a topological space $X$ is a maximal connected subspace of $X$
• Every point in a topological space belongs to exactly one connected component
• Connected components form a partition of the space
  • They are disjoint and their union is the entire space
• The connected components of a space can be used to study its global structure and properties

Locally connected spaces

• A topological space $X$ is locally connected if for every point $x \in X$ and every neighborhood $U$ of $x$, there exists a connected neighborhood $V$ of $x$ such that $V \subseteq U$
• In a locally connected space, small neighborhoods around each point are connected
• Locally connected spaces have nice properties, such as the equivalence of connectedness and path connectedness
• Many important spaces in noncommutative geometry, such as manifolds, are locally connected

Totally disconnected spaces

• A topological space $X$ is totally disconnected if its only connected subspaces are single points
• In a totally disconnected space, there are no non-trivial connected subsets
• Examples of totally disconnected spaces include the Cantor set and the $p$-adic numbers
• Totally disconnected spaces play a role in the study of ultrametric spaces and noncommutative geometry

Connectedness in product spaces

• The product of connected spaces is connected
  • If $X$ and $Y$ are connected spaces, then their product $X \times Y$ is also connected
• However, the converse is not true
  • A product space can be connected even if one or both of its factors are disconnected (line with two origins)
• Connectedness in product spaces is important when studying noncommutative spaces that arise as products or fibrations

Connectedness in quotient spaces

• If $X$ is a topological space and $\sim$ is an equivalence relation on $X$, the quotient space $X/\sim$ is the space of equivalence classes with the quotient topology
• If $X$ is connected and the equivalence relation $\sim$ is "compatible" with the topology (e.g., closed equivalence relation), then the quotient space $X/\sim$ is also connected
• Quotient spaces are fundamental in noncommutative geometry, as they allow for the construction of new spaces with desired properties

Connectedness in topological groups

• A topological group is a group $G$ equipped with a topology such that the group operations (multiplication and inversion) are continuous
• In a topological group, connectedness is related to the structure of subgroups
  • The connected component of the identity element is a normal subgroup
  • The quotient of a topological group by the connected component of its identity is a totally disconnected group
• Topological groups, especially Lie groups, play a crucial role in noncommutative geometry and gauge theory

Connectedness in metric spaces

• In a metric space $(X, d)$, connectedness can be characterized using sequences
  • A metric space is connected if and only if every Cauchy sequence converges
• Connectedness in metric spaces is related to completeness
  • Every connected complete metric space is path connected
• Many important spaces in noncommutative geometry, such as Riemannian manifolds and noncommutative tori, are metric spaces

Connectedness in Hausdorff spaces

• A topological space $X$ is Hausdorff if for any two distinct points $x, y \in X$, there exist disjoint open sets $U, V$ such that $x \in U$ and $y \in V$
• In a Hausdorff space, connected components are closed
  • This allows for a better understanding of the global structure of the space
• Many spaces studied in noncommutative geometry, such as smooth manifolds and noncommutative tori, are Hausdorff

Connectedness in compact spaces

• A topological space $X$ is compact if every open cover of $X$ has a finite subcover
• In a compact Hausdorff space, connectedness is equivalent to path connectedness
• Compact spaces have nice properties, such as the existence of maximal connected subspaces
• Compactness is important in noncommutative geometry, as it allows for the use of powerful analytical tools and the study of C*-algebras

Connectedness of real line

• The real line $\mathbb{R}$ with the standard topology is connected
  • It cannot be divided into two disjoint non-empty open sets
• The real line is also path connected
  • Any two points can be joined by a continuous path (line segment)
• The connectedness of the real line is fundamental in analysis and topology

Connectedness of complex plane

• The complex plane $\mathbb{C}$ with the standard topology is connected
  • It cannot be divided into two disjoint non-empty open sets
• The complex plane is also path connected
  • Any two points can be joined by a continuous path (line segment or curve)
• The connectedness of the complex plane is crucial in complex analysis and algebraic geometry

Applications of connectedness

• Connectedness is used in the study of manifolds and their properties
  • Connected manifolds have a well-defined dimension and orientability
• In noncommutative geometry, connectedness is important for understanding the global structure of noncommutative spaces
  • Connectedness of C*-algebras and their spectra
  • Connectedness of quantum groups and their homogeneous spaces
• Connectedness is also applied in other areas of mathematics, such as algebraic topology and dynamical systems