Tag Archives: topology

Topology

Continuous maps that are not closed or not open

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We recall some definitions on open and closed maps. In topology an open map is a function between two topological spaces which maps open sets to open sets. Likewise, a closed map is a function which maps closed sets to closed sets.

For a continuous function f:X↦Y, the preimage f−1(V) of every open set V⊆Y is an open set which is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X. However, a continuous function might not be an open map or a closed map as we prove in following counterexamples. Continue reading Continuous maps that are not closed or not open →

Topology

A curve filling a square – Lebesgue example

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Introduction

We aim at defining a continuous function φ:[0,1]→[0,1]2. At first sight this looks quite strange.

Indeed, φ cannot be a bijection. If φ would be bijective, it would also be an homeomorphism as a continuous bijective function from a compact space to a Haussdorff space is an homeomorphism. But an homeomorphism preserves connectedness and [0,1]∖{1/2} is not connected while [0,1]2∖{φ(1/2)} is.

Nor can φ be piecewise continuously differentiable as the Lebesgue measure of φ([0,1]) would be equal to 0.

φ is defined in two steps using the Cantor space K. Continue reading A curve filling a square – Lebesgue example →

Topology

Cantor set: a null set having the cardinality of the continuum

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Definition of the Cantor set

The Cantor ternary set (named Cantor set below) K is a subset of the real segment I=[0,1]. It is built by induction:

  • Starting with K0=I
  • If Kn is a finite disjoint union of segments Kn=∪k[ak,bk], Kn+1=⋃k([ak,ak+bk−ak3]∪[ak+2bk−ak3,bk])

And finally K=⋂n∈NKn. The Cantor set is created by repeatedly deleting the open middle third of a set of line segments starting with the segment I.

The Cantor set is a closed set as it is an intersection of closed sets. Continue reading Cantor set: a null set having the cardinality of the continuum →

Topology

An unbounded convex not containing a ray

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We consider a normed vector space E over the field of the reals R and a convex subset C⊂E.

We suppose that 0∈C and that C is unbounded, i.e. there exists points in C at distance as big as we wish from 0.

The following question arises: “does C contains a ray?”. It turns out that the answer depends on the dimension of the space E. If E is of finite dimension, then C always contains a ray, while if E is of infinite dimension C may not contain a ray. Continue reading An unbounded convex not containing a ray →

Topology

A compact whose convex hull is not compact

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We consider a topological vector space E over the field of the reals R. The convex hull of a subset X⊂E is the smallest convex set that contains X.

The convex hull may also be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.

The convex hull of X is written as Conv(X). Continue reading A compact whose convex hull is not compact →

Topology

A compact convex set whose extreme points set is not close

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Let’s remind that an extreme point c of a convex set C in a real vector space E is a point in C which does not lie in any open line segment joining two points of C.

The specific case of dimension 2

Proposition: when C is closed and its dimension is equal to 2, the set ˆC of its extreme points is closed.
Continue reading A compact convex set whose extreme points set is not close →

Topology

An empty intersection of nested closed convex subsets in a Banach space

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We consider a decreasing sequence (Cn)n∈N of non empty closed convex subsets of a Banach space E.

If the convex subsets are closed balls, their intersection is not empty. To see this let xn be the center and rn>0 the radius of the ball Cn. For 0≤n<m we have ‖xm−xn‖≤rn–rm which proves that (xn)n∈N is a Cauchy sequence. As the space E is Banach, (xn)n∈N converges to a limit x and x∈⋂+∞n=0Cn. Continue reading An empty intersection of nested closed convex subsets in a Banach space →