Topology

Stone–Čech compactification One of the proof of the Stone–Čech compactification is to consider the continuous functions from a given topological space $X$ to $[0, 1]$. Crux 1: constructing $[0, 1]^{C}$ Let $C$ be the space of all continuous functions from $X$ to $[0, 1]$, consider $[0, 1]^{C}$, there is a natural map from $X$ to $[0, 1]^{C}$: for each $x \in X$, define $\phi(x) = f \mapsto f(x)$. With product topology $\phi$ is continuous....

I was going through a problem in Lee’s Topological Manifolds book1 (problem 7-12). It states that the infinite broom has a strong deformation retract to $(0, 0)$ but not for $(1, 0)$ (only a deformation retract). Below is some of my playful observations inspired by this problem. (Not restricted to the infinite broom space.) Inifinite Broom Note that the deformation retract to $(0, 0)$ itselfs imply the deformation retract to $(1, 0)$, since one can define the deformation retract to $(1, 0)$ by “first retract to $(0, 0)$, then push it to the point $(1, 0)$ along the line joining these two points”....

Defining the machine by describing it In Sipser’s Introduction to the theory of computation, a alternative way of defining a Turing machine (other than defining the formal 7-tuple, which is a PITA) - by its “description”. Example (from the proof of $A_{TM}$ is deciable): M = “On input <B, w>, where B is a DFA and w is a string: 1. Simulate B on input w. 2. If the simulation ends in an accept state, accept ....