Considered the infinite broom

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Considered the infinite broom pictured here:
(a) A space is locally connected at r if for every neighborhood U of r, there is a
connected neighborhood V of z with V C U. Argue that the infinite broom is
not locally connected at p.
(b) A space is weakly locally connected at a if for every neighborhood U of r, there
is a connected subspace Y of X contained in U that contains a neighborhood V
of r. (So r e V C YC U.) Argue that the infinite broom is weakly locally
connected at p.
There is no exact definition of the space" so for simplicity (or not), declare that
d(p, am) = and that the height of the broom above a, is . Notice that this
means (why?) that any open ball centered at p containing a, contains the highest
branch of the broom above a,.
Transcribed Image Text:Considered the infinite broom pictured here: (a) A space is locally connected at r if for every neighborhood U of r, there is a connected neighborhood V of z with V C U. Argue that the infinite broom is not locally connected at p. (b) A space is weakly locally connected at a if for every neighborhood U of r, there is a connected subspace Y of X contained in U that contains a neighborhood V of r. (So r e V C YC U.) Argue that the infinite broom is weakly locally connected at p. There is no exact definition of the space" so for simplicity (or not), declare that d(p, am) = and that the height of the broom above a, is . Notice that this means (why?) that any open ball centered at p containing a, contains the highest branch of the broom above a,.
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