# January 2016

# Necessary vs. Sufficient?

In sum, the **sufficient condition** (a.k.a. the "if" direction) allows you to get what you want. That is, if you assume the sufficient condition, you'll obtain your desired conclusion. It's enough. *It's sufficient.*

On the other hand, the **necessary condition** (a.k.a. the "only if" direction) is the one you *must *assume in order to get what you want. In other words, if you don't have the necessary condition then you can't reach your desired conclusion. *It** is necessary*.

# Absolute Continuity (Part Two)

There are two definitions of absolute continuity out there. One refers to an absolutely continuous *function *and the other to an absolutely continuous *measure.* And although the definitions appear unrelated, they are in fact *very much* related, linked together by Lebesgue's Fundamental Theorem of Calculus. This is the second of a two-part series where we explore that relationship.

# Absolute Continuity (Part One)

There are two definitions of absolute continuity out there. One refers to an absolutely continuous *function *and the other to an absolutely continuous *measure*. And although the definitions appear unrelated, they are in fact *very much* related, linked together by Lebesgue's Fundamental Theorem of Calculus. This is part one of a two-part series where we explore that relationship.

# The Fundamental Group of the Circle, Part 6

Welcome to the final post in a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. Today we prove two lemmas (the path- and homotopy-lifting properties) that were used in parts four and five. The proof follows that found in Hatcher's Algebraic Topology section 1.1.