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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 2

Consider the space \(\mathcal {L}\) of all straight lines \(ax+by+c=0\) in the Euclidean plane \(\mathbb {R}^2\). Topologize \(\mathcal {L}\) so that a sequence of lines \(L_i\) is convergent to a line \(L=\{ax+by+c=0\}\) if and only if \(L_i\) can be represented by equations \(a_ix+b_iy+c_i=0\) so that \(\lim _i a_i=a\), \(\lim _i b_i=b\), and \(\lim _i c_i=c\). Prove that the topological space \(\mathcal {L}\) is homeomorphic to the Moebius band without boundary.

Raw LaTeX

Consider the space \(\mathcal{L}\) of all straight lines
\(ax+by+c=0\) in the Euclidean plane \(\mathbb{R}^2\). Topologize \(\mathcal{L}\) so that a
sequence of lines \(L_i\) is convergent to a line
\(L=\{ax+by+c=0\}\) if and only if \(L_i\) can be represented by equations
\(a_ix+b_iy+c_i=0\) so that \(\lim_i a_i=a\), \(\lim_i b_i=b\), and \(\lim_i c_i=c\).
Prove that the topological space \(\mathcal{L}\) is homeomorphic to the Moebius band
without boundary.

Topic Tags

Mayer-Vietoris Sequences Point-Set Topology and Separation Theorems Relative Homology and Long Exact Sequences Smooth Manifolds and Embedded Submanifolds