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  • Many students have the false belief that if a topological space is totally disconnected, then it must be discrete (related to examples already given). The rationals are a simple counter-example of course.

  • It is common to imagine rotation in an n-dimensional space, as a rotation through an "axis". this is of course true only in 3D. , In higher dimensions there is no "axis". but a n-2 dimensional subspace instead.

  • In calculus, I had some troubles with the following wrong idea. A curve in a plane parametrized by a smooth function is "smooth" in the intuitive sense (having no corners). the curve (-t^2,t^2) that is defined by $(t^2,t^2)$ for $t\ge0$ and $(-t^2,t^2)$ for $t<0$ is the graph of the absolute value function with a "corner" at the origin, though the coordinate functions t^2 and -t^2 are smooth. the "non-regularity" of the parametrization resolves the conflict.

  • When first encountering the concept of a spectrum of a ring, the belief that a continuous function between the spectra of two rings must come from a ring homomorphism between the rings.
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  • Many students have the false belief that if a topological space is totally disconnected, then it must be discrete (related to examples already given). The rationals are a simple counter-example of course.

  • It is common to imagine rotation in an n-dimensional space, as a rotation through an "axis". this is of course true only in 3D. In higher dimensions there is no "axis". but a n-2 dimensional subspace instead.

  • In calculus, I had some troubles with the following wrong idea. A curve in a plain plane parametrized by a smooth function is "smooth" in the intuitive sense (having no corners). the curve (-t^2,t^2) is the graph of the absolute value function with a "corner" at the origin, though the functions t^2 and -t^2 are smooth. the "non-regularity" of the parametrization resolves the conflict.

  • When first encountering the concept of a spectrum of a ring, the belief that a continuous function between the spectra of two rings must come from a ring homomorphism between the rings.
show/hide this revision's text 2 Removed the reference to n-2 dim'l subspace

Many students have the false belief that if a topological space is totally disconnected, then it must be discrete (related to examples already given). The rationals are a simple counter-example of course.

It is common to imagine rotation in an n-dimensional space, as a rotation through an "axis". this is of course true only in 3D. In higher dimensions there is no "axis" axis". but a n-2 dimensional subspace instead.

In calculus, I had some troubles with the following wrong idea. A curve in a plain parametrized by a smooth function is "smooth" in the intuitive sense (having no corners). the curve (-t^2,t^2) is the graph of the absolute value function with a "corner" at the origin, though the functions t^2 and -t^2 are smooth. the "noon-regularity" non-regularity" of the parametrization resolves the conflict.

When first encountering the concept of a spectrum of a ring, the belief that a continues continuous function between the spectra of two rings must come from a ring homomorphism between the rings.
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