Connectedness

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✓ 22 results · all verified · 13 also independently AI-judged

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1 · Prerequisites

Abstract Algebra

Linear Algebra

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua

Definition

Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

Theorem

For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant

Lemma

A subspace A ⊆ X is disconnected exactly when A = A₁ ∪ A₂ with A₁, A₂ nonempty and separated in X, which is the criterion this library already uses on the real line

Corollary

The connected subspaces of ℝ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ℝ"

Theorem

A continuous image of a connected space is connected, and connectedness is a topological property

Corollary

A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values

Theorem

A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member

Theorem

If A is connected and A ⊆ B ⊆ Ā then B is connected; in particular the closure of a connected set is connected

Theorem

A product of connected spaces is connected in the product topology, and that argument is a theorem of ZF; for an infinite index set it is the assertion that the product of nonempty spaces is nonempty that uses the Axiom of Choice

Definition

Connected components, quasicomponents, and totally disconnected spaces

Theorem

The components of a space are its maximal connected subsets, they partition it, and each of them is closed

Theorem

Every quasicomponent is a closed union of components, so each component is contained in a quasicomponent, and the quasicomponents partition the space

Definition

Paths, path-connected spaces and path components

Definition

Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point

Theorem

A space is locally connected exactly when every component of every open subspace is open; in that case the components of the space itself are clopen

Theorem

The long ray is a linear continuum, hence connected; every one of its at most countable subsets is bounded above, assuming countable choice

Lemma

The graph of the piecewise-linear map oscillating between 0 and 1 on the intervals [1/(n+2), 1/(n+1)] is path-connected, its closure adds the segment {0} × [0,1], and that closure is connected, is not path-connected because no path joins the segment to the graph, and is not locally connected

Remark

Which conventions this page fixes: the empty space and the one-point space, separated sets against disjoint open sets, and what is not developed here

Example

Connectedness

3 · Items

The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua

Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant

A subspace A ⊆ X is disconnected exactly when A = A₁ ∪ A₂ with A₁, A₂ nonempty and separated in X, which is the criterion this library already uses on the real line

The connected subspaces of ℝ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ℝ"

A continuous image of a connected space is connected, and connectedness is a topological property

A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values

A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member

If A is connected and A ⊆ B ⊆ Ā then B is connected; in particular the closure of a connected set is connected

A product of connected spaces is connected in the product topology, and that argument is a theorem of ZF; for an infinite index set it is the assertion that the product of nonempty spaces is nonempty that uses the Axiom of Choice

Connected components, quasicomponents, and totally disconnected spaces

The components of a space are its maximal connected subsets, they partition it, and each of them is closed

Every quasicomponent is a closed union of components, so each component is contained in a quasicomponent, and the quasicomponents partition the space

Paths, path-connected spaces and path components

Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point

A space is locally connected exactly when every component of every open subspace is open; in that case the components of the space itself are clopen

Every path-connected space is connected, and every path component lies inside a component

A connected, locally path-connected space is path-connected, because its path components are open

A linear continuum is connected in its order topology, and so is every order-convex subset of it

The closed long ray ω₁ × [0,1) under the lexicographic order, and the long line, with the order topology

The long ray is a linear continuum, hence connected; every one of its at most countable subsets is bounded above, assuming countable choice

The graph of the piecewise-linear map oscillating between 0 and 1 on the intervals [1/(n+2), 1/(n+1)] is path-connected, its closure adds the segment {0} × [0,1], and that closure is connected, is not path-connected because no path joins the segment to the graph, and is not locally connected

Which conventions this page fixes: the empty space and the one-point space, separated sets against disjoint open sets, and what is not developed here

Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected

FALSE: every connected topological space is path-connected

FALSE: the closure of a path-connected subspace is path-connected

FALSE: every connected space is locally connected

FALSE: the intersection of two connected subspaces is connected

FALSE: a totally disconnected space carries the discrete topology