To determine if there exists a bijection ( f: \mathbb{R}^n \to \mathbb{R}^n ) such that the forward map (induced on power sets) is connected but the inverse map is not, we analyze the properties of connectedness-preserving bijections.

Key Observations:

1. Connectedness-Preserving Maps: A map ( f ) is connected if it maps every connected subset ( S \subset \mathbb{R}^n ) to a connect...

Short answer:

• In dimension 1 the answer is yes: a bijection f: R → R that sends every connected subset to a connected subset must be monotone, hence continuous, hence a homeomorphism; therefore f−1 also preserves connectedness.
• In dimensions n ≥ 2 the answer is no: there exist bijections f: R^n → R^n that send every connected subset to a connected subset, while f−1 does not preserve connectedn...

This is a fascinating topological question about whether bijections of that preserve connectedness must have inverses that also preserve connectedness.

Let me analyze the key aspects:

The Question's Core We're looking for a bijection such that:

• maps connected sets to connected sets
• maps some connected set to a disconnected se...

At the moment nobody knows the answer in dimensions ≥ 2.
What is known is the following.

1. One–dimensional case
   If f : ℝ → ℝ is a bijection such that every connected set is carried to a connected set, then f is necessarily monotone. (If it were not monotone, there would be three points a<b<c with f(b) outside the closed interval [f(a),f(c)], and the image of the interval [a,c] ...

This is an excellent and deep question in point-set topology. The answer is no, such a bijection does not exist.

Theorem: Let be a bijection. If maps connected sets to connected sets, then its inverse also maps connected sets to connected sets.

The reason for this is a powerful and non-trivial theorem in topology:

Main Result: For any $n ...

<think> We are considering a bijection \( f: \mathbb{R}^n \to \mathbb{R}^n \) such that for every connected subset \( S \subseteq \mathbb{R}^n \), the image \( f(S) \) is connected. The question is whether the inverse map \( f^{-1} \) must also preserve connectedness, i.e., for every connected subset \( T \subseteq \mathbb{R}^n \), the preimage \( f^{-1}(T) \) is connected.

First, note that since...

Here is what is known (or not known) about the question

“Let f∶ℝⁿ→ℝⁿ be a bijection which has the property

 (⋆)  whenever S⊂ℝⁿ is (topologically) connected, so is f(S).

Must its inverse f⁻¹ also carry every connected set to a connected set?”

1. The one‐dimensional case is classical — a bijection of ℝ which preserves connected sets is automatically monotone, hence continuous, hence a...

Bijections that Preserve Connectedness and Their Inverses

This is an intriguing question about the relationship between a bijection that preserves connectedness and its inverse.

Understanding the Question

We're considering bijections with the standard topology, where maps connected sets to connected sets. The question asks whether the inverse map $f^{...

No.
For a bijection of Euclidean space the property  “every connected set has a connected image’’
is automatically inherited by the inverse map.
In fact a bijection (f:\mathbb R ^n\longrightarrow\mathbb R ^n) with that property is forced to be a homeomorphism, and a homeomorphism obviously carries connected sets to connected sets in both directions.

We prove the three required facts. ...