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Can the definition of The Long Line be clarified? A better (IMO) description of the long line is $[0,\Omega) \times [0,1)$ ordered lexicographically: $$(\alpha, t) \le_L (\beta, u) \iff (\alpha < \beta) \lor \left(\alpha=\beta \land (t \le u)\right)$$ and then given the order topology (with basic elements all open intervals plus all right-open intervals of the form $[(0,0), (\alpha, t) \rangle$ (special case for the minimum, there is no maximum)
general topology - Prove the long line is not contractible . . . Define the long ray to be the ordered set $\omega_1 \times [0,1)$ taken in lexicographical order As a space, it is given the order topology Define the long line to be the space obtained by gluing together two long rays together at their initial points Prove the long line is not contractible An outline of a proof can be given as follows:
general topology - Regarding the connectedness of the Long Line . . . First, let's talk about what the long line is (n't) You write: I believe that the Long Line is a mapping from the real line to the real line such that each point on the Long Line represents a copy of the real line This doesn't quite parse for me, but it sounds like you're talking about a particular topology on $\mathbb {R}\times\mathbb {R
general topology - The Long Line is not second countable - Mathematics . . . Then (α + 1, 1 2) (α + 1, 1 2) has a neighbourhood {α + 1} × (0, 1) {α + 1} × (0, 1) in L L that misses all members of A A This shows that no countable subset of L L can be dense, i e L L is not separable As a second countable space is always separable, L L does not have a countable base as well Share
general topology - Prove long line is path-connected. - Mathematics . . . $\begingroup$ Also see this note, of which you should only not trust the last statement: the long line is normal (despite being non-paracompact, but it is countably paracompact) The fact that the initial segments are homeomorphic to open intervals in $\Bbb R$, as shown, implies path-connectedness
general topology - Is there such a thing as the *longer* line . . . This is in reference to the long ray (Alexandroff line) which can be extended in both directions to form the so-called Long Line defined by: L1 = {ω1 × [0, 1)} ∪ {ω1 × [−1, 0)} So the long plane would be defined as: L1 ×L1 A similar object is mentioned in the wikipedia article on the bagpipe theorem As for the "longer line" (or maybe
general topology - Prove that the long line is normal - Mathematics . . . The long line L L is the ordered set SΩ × [0, 1) S Ω × [0, 1) in the dictionary order, with its smallest element deleted And SΩ S Ω denotes the minimal uncountable well-ordered set I want to prove that L L is normal, but not metrizable To show, my attempt was to show that L L is linear continuum But it fails because L L is not linear
geometry - How to find coordinates of reflected point? - Mathematics . . . For finding the image of the point in the same line, we just multiply the rightmost term by 2 So, the image of the point (x1, y1) in the line ax1 + by1 + c = 0 is given by: x − x1 a = y − y1 b = − 2(ax1 + by1 + c) a2 + b2 The image of the point is at the same distance from the line as the point itself is from the line
What is the equation for a 3D line? - Mathematics Stack Exchange 1 Besides the parametric form, another equation of a line in 3D to get it in the form f(x, y, z) = 0 could be written as: r − r0 r − r0 ⋅ n = 1 Here r = (x, y, z) is a vector representing any general point on the line r0 = (x0, y0, z0) is a given point that lies on the line n = (nx, ny, nz) is a given unit vector (that has a magnitude