In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also referred to as the broom space. [1]
The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the origin to the point (1, 1/n) as n varies over all positive integers, together with the interval (½, 1] on the x-axis. [2]
The closed infinite broom is then the infinite broom together with the interval (0, ½] on the x-axis. In other words, it consists of all closed line segments joining the origin to the point (1, 1/n) or to the point (1, 0). [2]
Both the infinite broom and its closure are connected, as every open set in the plane which contains the segment on the x-axis must intersect slanted segments. Neither are locally connected. Despite the closed infinite broom being arc connected, the standard infinite broom is not path connected. [2]
The interval [0,1] on the x-axis is a deformation retract of the closed infinite broom, but it is not a strong deformation retract.
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "holes" or "missing endpoints", i.e. that the space not exclude any "limiting values" of points. For example, the "unclosed" interval (0,1) would not be compact because it excludes the "limiting values" of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers ℚ is not compact because it has infinitely many "holes" corresponding to the irrational numbers, and the space of real numbers ℝ is not compact either because it excludes the limiting values and . However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in Euclidean space, but may be inequivalent in other topological spaces.
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