Connectedness and paths

Try

A continuous real-valued function on an interval cannot skip an intermediate output. Connectedness generalizes the domain property behind that statement. A space is connected when it cannot be split into two disjoint nonempty relatively open subsets whose union is the whole space. This definition concerns global separation, not how close individual points look in a diagram.

Predict

Predict whether a finite two-point space is connected. In any metric, two distinct points have positive distance, so sufficiently small balls separate them. It is compact but disconnected. An interval can be connected without being compact, as R itself shows. These properties answer different questions and should not be treated as progressively stronger versions of one vague geometric idea.

Experiment

Compare examples.

Space	Connected?	Compact?
[0,1]	Yes	Yes
R	Yes	No
Two distinct points	No	Yes

Draw a separation for the last row and explain why overlapping open sets would not qualify. A separation requires disjoint pieces covering everything, both nonempty. Relative openness matters: the pieces need only be open within the space under discussion.

Understand

Every interval in R is connected. If an interval had a separation into U and V, choose a∈U and b∈V with a<b. Let s be the supremum of points of U in [a,b]. Relative openness rules out s belonging to U unless there are later U points, and rules out s belonging to V because U points approach it from below; endpoint cases are excluded using the neighborhoods of a and b. Since the interval contains s and U,V cover it, this is impossible. Conversely, a connected subset of R must be an interval: a missing point c between two of its points separates it by intersection with the two open half-lines around c.

Continuous images preserve connectedness. A separation of the image into two relatively open pieces would pull back to a separation of the domain. Nonemptiness of the image pieces ensures nonempty preimages. Therefore a continuous real-valued function on any connected space has an interval as its image, giving the intermediate-value property without coordinates in the source.

Apply

Path connectedness means every pair of points can be joined by a continuous map from [0,1] into the space with those endpoints. It implies connectedness: a separation would pull back along a path joining opposite pieces to separate [0,1]. The converse is false in general; proving a particular counterexample, such as the closure of the oscillating sine graph, requires careful topology and is not supplied merely by its picture here.

Repair “a connected set must contain the straight segment between its points.” That stronger property is convexity in a vector space. A circle is path connected but generally omits the chord between two points. In an arbitrary metric space, straight segments may not even be defined.

Continue

Optional depth proves closure preserves connectedness. Every nonempty relatively open part of the closure of A meets A. A separation of the closure would therefore restrict to two nonempty open pieces separating A, a contradiction. This argument uses density of A in its closure, not compactness.

Save one inverse-image proof and one real-line separation argument. Keep connectedness, path connectedness and convexity distinct in the conclusion. Next, contraction maps will use quantitative distances rather than topological connectedness to force one fixed point and produce explicit iterative error bounds.