Title: Theory / Unimportant / Math / Topology / wp/
Tags: récupéré fromMath topology mathToDef basis mathToSee

Page: , https://en.wikipedia.org/wiki/Lower_limit_topology
Name: The lower limit topology
Set: The real line
Chapter: Intro
Base: the intervals \([a,b)\)
Metric: none
Notation: \(\mathbb R_\ell\)
Locally compact: no
Compact: Its compact subsets are at most countable
Name2: Sorgenfrey Line
Paracompact: yes
Clopen: each \([a,b)\)
Limit points: any points, approached only from the right
Name3: The arrow

Page: xii
Set: closed unit sqare
Chapter: Intro
Notation: \(I_o^2\)
Order: dictionary order topology

Page: 077, https://en.wikipedia.org/wiki/Trivial_topology
Name: The trivial topology of \(X\)
Set: \(X\)
Chapter: 2
Section: 13
Variables: \(X\) a set
Kind: Ex
Base: \(\{X\}\)
Opens: \(\{X,\emptyset\}\).
Metric: \((x,y)\mapsto 0\)
Name2: Indiscrete space
Order: No
Regular: yes
Baire: Yes
Clopen: every set
Dense: Any non-empty set
Isolated points: none
Limit points: each point for each sequence (assuming at least 2 points in \(X\))
Closure: The closure of each non-empty set is \(X\)
T0/Kolmogorov: No (assuming at least 2 elements)

Page: 84, 86
Name: The ordered topology of \(X\)
Set: \(X\)
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
Base: -\((a,b)\)
-\((a,b]\) for \(b\) the maximal element of \(X\) and
-\([a,b)\) for \(a\) the minimal element of \(X\).
Prebase: Rays

Page: 88
Name: The subspace topology engendered by \(Y\)
Set: \(Y\)
Chapter: 2
Section: 16
Variables: \((X,\mathcal T)\) be a topology
\(Y\subseteq X\)
Kind: Definition
Base: \(\{Y\cap U\mid U\in\mathcal B\}\) for \(\mathcal B\) a basis of \(Y\)
Opens: \(\{Y\cap U\mid U\in\mathcal T\} \).
Notation: \(\mathcal T_Y\)

Page: https://en.wikipedia.org/wiki/Appert_topology
Name: Appert topology
Set: \(\mathbb N^{>0}\)
Opens: Sets which, either, do not contain 1, or asymptotically contains almost every positive integer
Locally compact: no (no compact neighborhood of 1)
Compact: its finite subsets
T5/completely normal Hausdorff: yes

Page: https://en.wikipedia.org/wiki/Appert_topology
Name: Arens-Fort space
Set: \(\mathbb N^2\)
Opens: Either:
-it does not contains \((0,0)\)
-it contains \((0,0)\) and all but a finite number of points of all but a finite number of columns
Metric: none
Compact: no
T4/normal Hausdorff: yes

Page: https://en.wikipedia.org/wiki/Rose_(topology)
Name: A rose
Set: \(n\) circles, with a point in common
Metric: standard metric of \(\mathbb R^2\)
Name2: A bouquet of \(n\) circles
Fundamental group: Free, with a generator for each petal.
Universal cover: An infinite tree, with edges labelled by positive/ negative and a circle

Page: https://en.wikipedia.org/wiki/Hedgehog_space
Name: The \(K\)-hedgehog space
Set: \(K\) copies of \((0,1]\), and a \(0\)
Variables: \(K\) some cardinal
Metric: Paris metric

Name: The Paris distance
Definition: For \(x,y\in V\), [$]d(x,y)=\norm{x-y}[/$] if \(0,x,y\) are on the same line, and [$]\norm{x}+\norm{y}[/$] otherwise.
Page: https://en.wikipedia.org/wiki/Hedgehog_spaced
Variables: \(V\) a normed vector space

Page: https://en.wikipedia.org/wiki/Tychonoff_plank
Name: Tychonoff plank
Set: \([0,\omega]\times[0,\omega_1]\)
Context: Récupéré
Locally compact: yes
Definition: Product of opens of the order topology
Completely normal regular: no because the deleted Tychonoff plank is not normal.
Perfectly normal T0: No because \(\{\infty\}\) is closed but not \(G_\delta\)

Page: https://en.wikipedia.org/wiki/Tychonoff_plank
Name: deleted Tychonoff plank
Set: \([0,\omega]\times[0,\omega_1]\setminus\{(\omega,\omega_1)\}\)
Opens: The opens of the Tychonoff plank, minus \((\omega,\omega_1)\)
Context: Récupéré
Isolated points: no

Page: https://en.wikipedia.org/wiki/Order_topology
Name: Right order topology
Set: \(X\)
Variables: \((X,<)\) a totally ordered space
Opens: \((a,\infty)\) for \(a\in X\)
Context: Récupéré
Locally compact: yes if bounded
Normal: no if bounded

Page: https://en.wikipedia.org/wiki/Order_topology#References
Opens: \([0,\omega_1]\)
Context: Récupéré
Locally compact: yes
Locally Hausdorff: no, because of \(\omega_1\)
T2 1/2/Urysohn: no

Page: https://en.wikipedia.org/wiki/Order_topology#References
Set: \(\alpha\)
Variables: \(\lambda\) an ordinal
Context: Récupéré
Notation: Standard order
Locally compact: iff \(\alpha\) is a successor ordinal
T4/normal Hausdorff: yes
Path connected: not in general
Assuming: The ordinal topology of \(\alpha\)
Dyadic: 0
Quasiregular: 0 and successors
Semiregular: the limit ordinals

Page: https://en.wikipedia.org/wiki/Order_topology#References
Context: Récupéré
Locally compact: yes
T2 1/2/Urysohn: no
Assuming: \(\omega_1\) topology

Name: An arc in \(X\)
Definition: An homeomorphism between \([0,1]\) and \(X\)
Page: https://en.wikipedia.org/wiki/Connected_space#Arc_connectedness
Variables: \((X,\mathcal T)\) a topological space
Extra: path ?

Name: An arc-component of \(X\)
Definition: A set of components which are pairwise connected by arcs
Page: https://en.wikipedia.org/wiki/Connected_space#Arc_connectedness
Variables: \((X,\mathcal T)\) a topological space

Name: An arc-connected set
Definition: A set with a single arc-component
Page: https://en.wikipedia.org/wiki/Connected_space#Arc_connectedness
Variables: \((X,\mathcal T)\) a topological space

Name: An H-space
Definition: \(\mathcal T\) a topological space on \(X\)
Page: https://en.wikipedia.org/wiki/H-space
Definition2: \(\cdot, e\) a continuous monoid on \(X\)
Example: Topological monoid
Conjdef: A tuple with \(X\) a set and

Name: A topological monoid
Definition: An H-space whose operation is associative
Page: https://en.wikipedia.org/wiki/Topological_monoid

Name: A topological group
Definition: A topological monoid whose operation admits an inverse, which is continuous
Page: https://en.wikipedia.org/wiki/Topological_group
Definition2: Group objects in the category of topology
Example: \((\mathbb K^n,+,0)\) with standard topology for \(\mathbb K\) being rationals, reals or complex
Name2: \(GL(n,R)\)
Abbreviation2: Orthogonalgroup

Name: A discrete group
Definition: A group with the discrete topology
Page: https://en.wikipedia.org/wiki/Topological_group
Variables: \(X\) a topological group,

Name: The set of topological groups
Page: https://en.wikipedia.org/wiki/Topological_group
Under: subgroups, for the subspace topology
Under2: set of left coset by a normal subgroup with quotient topology

Page: https://en.wikipedia.org/wiki/Tychonoff_plank
Name: Deleted Tychonoff plank
Set: \([0,\omega_1]\times[0,\omega]\setminus\{(\omega_1,\omega)\}\)
Chapter: 4
T4/normal Hausdorff: Not
Regular: Yes

Page: https://en.wikipedia.org/wiki/Tychonoff_plank
Name: Tychonoff plank
Set: \([0,\omega_1]\times[0,\omega]\)
Chapter: 4
Compact: Yes
Perfectly normal: Not because \(\{\infty\}\) is closed but not \(G_{\delta}\) set

Page: https://en.wikipedia.org/wiki/Hawaiian_earring
Set: set of circles of radius \(\frac1n\) and center \((0,\frac 1n)\)
Compact: yes
Fundamental group: Uncountable. Contains the free group of countably infinitely many generators as a proper subgroup. Contains also the loop which on \([2^{-n},2^{-n-1}]\) circumnavigates the nth circle.
Simply connected: not. the loop parametrising any circle is not homotopic to a trivial loop.
Semi-locally simply connected: not
Compactification of: one point compactification of a countable infinite family of open intervals