#homeomorphism

Return exactly the same text but decorate important words and names with markdown. Be creative. Use **bold**, *italic* and insert relevant **[links](https://en.wikipedia.org/wiki/Topology)**: **Topology** is a branch of mathematics that studies the properties of geometric objects that remain unchanged under certain transformations. It deals with concepts like **continuity**, **connectivity**, and…

Homeomorfismos pseudo-Anosov: Foliaciones medibles y rectángulos geometrizados.

I wanted to write again because I realised that, in my previous post I didn’t say anything about connectedness being a topological property, which is kind of the point!

What do I mean by this? The point is that if one topological space is connected, so is any other that is homeomorphic to it. We show this now.

Proposition. Connectedness is a Topological Invariant. That is, if a topological space is connected, so is any space homeomorphic to it.

Proof. Let \(A\) and \(B\) be two homeomorphic spaces. The above is equivalent to ‘if \(B\) isn’t connected, neither is \(A\)’. Suppose \(B\) isn’t connected, that is, we can write \(B\) as the union of two disjoint non-empty open sets, \(X\) and \(Y\). Let \( \phi: A \rightarrow B \) be the homeomorphism between \(A\) and \(B\). Since \(\phi\) is continuous, \(\phi^{-1}(X)\) and \(\phi^{-1}(Y)\) must be open in \(A\). Further, just because \(\phi\) is a function, \(\phi^{-1}(X)\) and \(\phi^{-1}(Y)\) must be disjoint. Finally, because \(\phi\) is surjective (onto), we must have \(A = X \cup Y\), meaning \(A\) is disconnected. \( \square \)

Notice that, in the proof, we only used the fact that \( \phi \) was a continuous mapping from \( A \) to \( B \) and that it was surjective. Hence, we have shown that, if \( A \) is connected, and we have a continuous surjective map \( \phi: A \rightarrow B \), then \( B \) must be connected. A problem with this version of things is that a continuous surjective mapping from a disconnected space, \(A\), to \(B\) does not guarantee that \(B\) is disconnected. Consider the following counterexample: let our topological spaces be \( A = (-1,-2) \cup (1,2) \) and \( B = (1,4) \), both with the induced topology from \(\mathbb{R}\). The following is a surjective continuous map from one to the other:

\begin{align*} f: A &\rightarrow B \\\\ x &\mapsto x^2, \end{align*}

but \( A \) is disconnected and \( B \) is connected.

Graph of \(f\)

The advantage of thinking about homeomorphisms is that they also preserve the property of being disconnected, which follows from the above proposition. In fact, they even preserve the number of connected components of a space, meaning, if a space has 5 connected components, so must any space homeomorphic to it. We show this now.

Proposition. Homeomorphisms preserve the number of connected components of a space. That is, if a space has \(n\) connected components, so does any other space homeomorphic to it.

Proof. Let \(A\) have \(n\) connected components and suppose \(A\) is homeomorphic to \(B\) with homeomorphism, \(\phi: A \rightarrow B\). Let the connected components of \(A\) be \( X_i \) for \( i \in \{1,...,n\} \), then each \( \phi ( X_i ) \) can be considered as a separate topological space (with induced topology) and must be connected, by our earlier proposition. They must also be disjoint, because \(\phi\) is a function. The question now becomes `are they the maximal connected subsets?’. The answer to this must be yes, because if there was some connected \( Y \supset \phi(X_i) \), then \(\phi^{-1}(Y)\) must be connected, by our earlier proposition. However, we have \( \phi^{-1} (Y) \supset X_i \), which can’t be connected in \(A\), since \(X_i\) is a maximal connected subset. Hence, we have a contradiction and the \(\phi(X_i)\) must be the maximal connected subsets of \(B\). Since, \(\phi\) is injective, there must be \(n\) of these. \(\square\)

You’ll be hard-pressed to turn away from this twisting, morphing projection-mapped dome in Santa Fe

(Courtesy Ouchhh)

Visually and aurally mesmerizing, a new 3D projection by Turkish design studio Ouchhh immerses the viewer in a psychedelic, eye-of-the-storm experience of whirling fractals inside a darkened dome.

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These two links have homeomorphic exteriors, that is, it is possible to transform the space around the first link into the second without cutting or glueing. Kenneth Baker presents the following series on his wonderful blog Sketches of Topology. Also check out his Flickr page.

I would give money to see an animated version of this.