Category Archives: Topology

Topology

A separable space that is not second-countable

Leave a comment

In topology, a second-countable space (also called a completely separable space) is a topological space having a countable base.

It is well known that a second-countable space is separable. For the proof consider a second-countable space \(X\) with countable basis \(\mathcal{B}=\{B_n; n \in \mathbb{N}\}\). We can assume without loss of generality that all the \(B_n\) are nonempty, as the empty ones can be discarded. Now, for each \(B_n\), pick any element \(b_n\). Let \(D=\{b_n;n \in \mathbb{N}\}\). \(D\) is countable. We claim that \(D\) is dense in \(X\). To see this let \(U\) be any nonempty open subset of \(X\). \(U\) contains some \(B_p\), hence \(b_p \in U\). So \(D\) intersects \(U\) proving that \(D\) is dense.

What about the converse? Is a separable space second-countable? The answer is negative and I present below a counterexample. Continue reading A separable space that is not second-countable →

Topology

A connected not locally connected space

1 Comment

In this article, I will describe a subset of the plane that is a connected space while not locally connected nor path connected.

Let’s consider the plane \(\mathbb{R}^2\) and the two subspaces:
\[A = \bigcup_{n \ge 1} [(0,0),(1,\frac{1}{n})] \text{ and } B = A \cup (\frac{1}{2},1]\] Where a segment noted \(|a,b|\) stands for the plane segment \(|(a,0),(b,0)|\). Continue reading A connected not locally connected space →

Topology

Continuous maps that are not closed or not open

Leave a comment

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 \(f: X \mapsto Y\), the preimage \(f^{-1}(V)\) of every open set \(V \subseteq Y\) 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

Leave a comment

Introduction

We aim at defining a continuous function \(\varphi : [0,1] \rightarrow [0,1]^2\). At first sight this looks quite strange.

Indeed, \(\varphi\) cannot be a bijection. If \(\varphi\) 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 \([0,1] \setminus \{1/2\}\) is not connected while \([0,1]^2 \setminus \{\varphi(1/2)\}\) is.

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

\(\varphi\) 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

2 Comments

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 \(K_0=I\)
  • If \(K_n\) is a finite disjoint union of segments \(K_n=\cup_k \left[a_k,b_k\right]\), \[K_{n+1}=\bigcup_k \left(\left[a_k,a_k+\frac{b_k-a_k}{3}\right] \cup \left[a_k+2\frac{b_k-a_k}{3},b_k\right]\right)\]

And finally \(K=\displaystyle \bigcap_{n \in \mathbb{N}} K_n\). 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

Leave a comment

We consider a normed vector space \(E\) over the field of the reals \(\mathbb{R}\) and a convex subset \(C \subset E\).

We suppose that \(0 \in C\) 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

Leave a comment

We consider a topological vector space \(E\) over the field of the reals \(\mathbb{R}\). The convex hull of a subset \(X \subset E\) 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 \(\mbox{Conv}(X)\). Continue reading A compact whose convex hull is not compact →

Topology

A compact convex set whose extreme points set is not close

Leave a comment

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 \(\hat{C}\) 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

Leave a comment

We consider a decreasing sequence \((C_n)_{n \in \mathbb{N}}\) 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 \(x_n\) be the center and \(r_n > 0\) the radius of the ball \(C_n\). For \(0 \leq n < m\) we have \(\Vert x_m-x_n\Vert \leq r_n – r_m\) which proves that \((x_n)_{n \in \mathbb{N}}\) is a Cauchy sequence. As the space \(E\) is Banach, \((x_n)_{n \in \mathbb{N}}\) converges to a limit \(x\) and \(x \in \bigcap_{n=0}^{+\infty} C_n\). Continue reading An empty intersection of nested closed convex subsets in a Banach space →