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Questions tagged [alexandroff-double-circle]

For questions about the Alexandroff double circle, also called "Concentric Circles" in Steen & Seebach's "Counterexamples in Topology".

The Alexandroff double circle is the space with underlying set $X = C_1 \cup C_2$ (where $C_i = \{ x \in \mathbb{R}^2 : |x| = i \}$), having the topology generated by the following sets

  • $\{ x \}$, where $|x| = 2$, and
  • $U \cup ( 2U \setminus F )$ where $U$ is a union of open arcs in $C_1$, $2U$ is the radial projection of $U$ onto $C_2$, and $F$ is a finite subset of $C_2$.

This space is example 97 in Steen and Seebach's Counterexamples in Topology, where it is called "Concentric Circles".

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Is the Alexandroff double circle compact and Hausdorff?

I recently encountered the Alexandroff double circle. The underlying set is $C = C_1 \cup C_2$, where $C_i$ is the circle of radius $i$ and centre $0$ in the complex plane. The basic open sets are: $\{z\}$ for every $z \in \mathbb{C}$ with $|z| =…
ThinkConnect
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My proof that the Alexandroff double circle not second-countable

I'm hoping someone can comment on if my logic on the Alexandroff double cirlce not being second countable is right. The Alexandroff double circle has underlying set $C = C_1 \cup C_2$ where $C_i = \{ z \in \mathbb{C} : |z| = i \}$ is the circle of…
ThinkConnect
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What do open sets look like in the Alexandroff double cirlce?

I recently encountered the following topological space, called the Alexandroff double cirlce: The underlying set is $C = C_1 \cup C_2$, where $C_i$ is the circle of radius $i$ and centre $0$ in the complex plane. Basic open sets are: $\{ z\}$ for…
ThinkConnect
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Example of $f$, $g$ and $a$, such that $(f \circ g)'(a)$ exists, and $g'(a)$ exists, but $f'(g(a))$ does not.

Find an example of 2 functions $f$ and $g$ and a point $a \in \mathbb{R}$, such that $(f \circ g)'(a)$ and $g'(a)$ exists, but $f'(g(a))$ does not exist, and also $f$ and $g$ must take on all values in $\mathbb{R}$. Same problem as this one. The…
user838035
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Find the centre of the circle

So there is a square of 50cm length. There are 3 circle's whose centre's are any 3 vertices of the square.The radius(r1,r2,r3) of these circles can be assumed i.e it is known. Now a 4th circle exists which lies on the 4th vertices and is touching…
Rachit Juthani
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Is the Alexandroff double circle separable?

Is the Alexandroff double circle separable (i.e. has a countable dense subset)? The Alexandroff double circle is the space with underlying set $C = C_1 \cup C_2$, where $C_i$ is the circle of radius $i$ and centre $0$ in the complex plane. The basic…
ThinkConnect
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Is the Alexandroff Double Circle first-countable?

The Alexandroff Double Circle is the topological space with underlying set $C = C_1 \cup C_2$, where $C_i$ is the circle of radius $i$ and centre $0$ in the complex plane. The basic open sets are: $\{ z\}$ for every $z$ with $|z| = 2$, and $U \cup…
ThinkConnect
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