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• Functions
  In mathematics, we refer to any loose collection of objects or entities -- of any nature -- as a ''set''. ...example, is this a circle in a bag of marbles? No, the marbles it is made of aren't connected to each other or to any location.
  151 KB (25,679 words) - 17:09, 20 February 2019
• Continuous functions
  ==Continuity as preservation of proximity== Let's review the stages our definition of continuity<!--\index{continuity}--> went through.
  42 KB (7,138 words) - 19:08, 28 November 2015
• Manifolds
  ==What is the topology of the physical Universe?== ...sun deviates from a straight line may be considered as evidence in support of this idea:
  51 KB (8,919 words) - 01:58, 30 November 2015
• Quotients
  A new way of building new things from old is ''gluing'': ...an build a lot of (topologically) different things with nothing but sheets of paper and a glue-stick:
  26 KB (4,538 words) - 23:15, 26 November 2015
• Compact spaces
  ...ues between $f(a)$ and $f(b)$. It follows from this theorem that the image of a path-connected space<!--\index{path-connectedness}--> (under a continuous '''Exercise.''' Generalize the theorem to the case of $f:[a,b]\cup [c,d] \to {\bf R}$.
  19 KB (3,207 words) - 13:06, 29 November 2015
• Functions of several variables OLD
  Before we consider [[continuity]] of functions of several variables, let's recall how the issue is handled in [[Calc1]] as th ...us. However it is right-continuous at $\{0 \}$ as $x = 0$ is the end point of $D(f)$.
  34 KB (5,636 words) - 23:52, 7 October 2017
• Linear functions in Euclidean space
  ...ces on the left and then, on the right, they are interpreted as the graphs of two functions. Let's look at the properties of a function of this kind.
  23 KB (3,893 words) - 04:43, 15 February 2013
• Topology Illustrated -- Index
  *[[basis of neighborhoods|basis of neighborhoods]] *[[basis of topology|basis of topology]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Introductory algebraic topology: review exercises
  *[[Intersection of any collection of closed sets is closed ]] *[[Is a closed subset of a compact space always compact? ]]
  9 KB (1,553 words) - 20:10, 23 October 2012
• Constructions
  ...can build a lot of topologically different things with nothing but sheets of paper and a glue-stick: <!--150-->[[image:glue edges of paper.png| center]]
  13 KB (2,270 words) - 22:14, 18 February 2016
• More about manifolds
  ==Quotients of manifolds== ..., i.e., it's a $1$-dimensional [[manifold]]. Now we want to see the effect of [[gluing]] on its structure.
  9 KB (1,542 words) - 19:58, 21 January 2014
• Point-set topology: exercises
  #Prove that the space of continuous functions $f:[0,1]\rightarrow {\bf R}$ is a metric space with th #Is the boundary of a set always closed?
  5 KB (814 words) - 16:40, 4 October 2013
• Compactness
  <TR> <TD>If you are looking for "compactness" as a measure of [[roundness]] of objects, then go there... </TD></TR> ...e. Then at least on of the intervals will contain infinitely many elements of the sequence. Then you go on to prove convergence.
  4 KB (635 words) - 12:57, 12 August 2015
• Chain complexes of cell complexes
  The ''spaces'' that we study will be made of elementary pieces with "trivial" [[topology]]. As an example, a [[cubical complex]] has been understood as a collection of cubical cells on the grid ${\bf Z}^n$ so that the [[boundary operator]] is
  7 KB (1,225 words) - 14:05, 4 August 2013
• Linear operators: part 1
  Since a lot of properties of [[linear operator]]s have nothing to do with [[linearity]], we start with t ==Compositions and inverses, of functions==
  13 KB (2,086 words) - 19:58, 27 January 2013
• Homeomorphism
  ...}--> can appear. If we choose to think of continuous functions as mappings of the terrain, we are trying to learn how to answer the most basic question: The continuity of the mapping prevents it from tearing the map and guarantees that we will ne
  13 KB (2,168 words) - 13:09, 7 August 2014
• Path-connectedness
  ...→ Y$ is [[continuous]]. Then, if $X$ path-connected then so is its [[image of function|image]], $f(X)$, under $f$. ...ow that connectedness is a [[topological invariant]]. And so is the number of components, see [[Betti numbers]].
  8 KB (1,315 words) - 13:15, 12 August 2015
• Short overview
  The ''spaces'' that we study will be made of elementary pieces with "trivial" [[topology]]. As an example, a [[cubical complex]] has been understood as a collection of cubical cells on the grid ${\bf Z}^n$ so that the [[boundary operator]] is
  7 KB (1,179 words) - 15:27, 7 January 2014
• Preimage
  ...ion $f: X → Y$ and a point $b$ in $Y$, the ''preimage'' of $b$ is a subset of $X$: An alternative terminology is that $f^{-1}(b)$ are the ''fibers'' of $f$.
  576 bytes (106 words) - 14:04, 1 November 2012
• Introduction to abstract mathematics: course
  *[[Rules of proof writing]] **examples and types of proofs:
  3 KB (373 words) - 16:06, 25 September 2013

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