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I have tried to prove that $\mathbb{Q}\subset \mathbb{R}$ equipped with the subspace topology is a disconnected space. I'd like to make sure my attempted proof is correct since topological properties of $\mathbb{Q}$ seem like nice things to be familiar with.
First, a base for the subspace topol...user84215
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In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment. Cantor himself mentioned the ternary construction only in...
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In mathematics, arg is a function operating on complex numbers (visualized in a complex plane). It gives the angle between the positive real axis to the line joining the point to the origin, shown as φ in figure 1, known as an argument of the point.
== Definition ==
An argument of the complex number z = x + iy, denoted arg(z), is defined in two equivalent ways:
Geometrically, in the complex plane, as the angle φ from the positive real axis to the vector representing z. The numeric value is given by the angle in radians and is positive if measured counterclockwise.
Algebraically, as any r...
In the text "An Introduction to Measure and Integration by Rana" i'm having trouble gaining intuition behind the following Proposition in $(1)$
$(1)$
$$\text{Proposition}$$
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,\, ...
