<we><edit> <!-- questions and answers -->
optimization
hilbert-spaces
metric-spaces
proof-verification
combinatorics
finite-groups
induction
logarithms
number-theory
real-analysis
continuity
reference-request
ring-theory
prime-numbers
pde
general-topology
arithmetic
linear-algebra
convex-analysis
group-theory
euclidean-geometry
multivariable-calculus
taylor-expansion
functional-analysis
lebesgue-integral
general-topology
Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$
general-topology
algebraic-topology
homotopy-theory
fundamental-groups
Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$.
general-topology
algebraic-topology
homotopy-theory
fundamental-groups
Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.
general-topology
algebraic-topology
homotopy-theory
fundamental-groups
free-product
Unions of topological dynamical systems - a counter example
general-topology
dynamical-systems
examples-counterexamples
Does every torus $T\subset S^3$ bounds a solid tours $S^1\times D^2\subset S^3$?
general-topology
manifolds
geometric-topology
Does set with Lebesgue-Mass nonzero have almost surely an open subset
general-topology
measure-theory
lebesgue-measure
borel-sets
Why are continuous functions the "right" morphisms between topological spaces?
general-topology
soft-question
category-theory
Is below subspace is connected?
real-analysis
general-topology
connectedness
Is the converse of Tychonoff's theorem true?
general-topology
Is the following embedding of $k5$ on torus not a 2-cell embedding? If not, how can we get the 2-cell embedding of $k5$ on the torus?
general-topology
graph-theory
Definition of disconnected?
real-analysis
general-topology
connectedness
About spaces $\Bbb R^n/\!\sim$ in which certain coordinate permutations do not matter.
general-topology
permutations
equivalence-relations
symmetric-groups
quotient-spaces
Prove isomorphism in infinite-dim condition
linear-algebra
abstract-algebra
general-topology
Action of universal covering deck transformations group
general-topology
algebraic-topology
some confusion in cofinite topology..
general-topology
Convergence in a Topological Space
general-topology
convergence
Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$
general-topology
order-topology
Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$.
general-topology
algebraic-topology
homotopy-theory
fundamental-groups
path-connected
Definition of topology induced by seminorms
general-topology
topological-vector-spaces
Can we recover a topology from knowing about continuity of maps from compact spaces?
general-topology
$X$ compact convex subset of $\mathbb{R^n}$. $f:X\to X,f(Fr(X))\subset Fr(X)$. $f|_{Fr(X)}$ is homotopic with identity on $Fr(X)$. Show that $f(X)=X$
general-topology
homotopy-theory
The Weak topology on an infinite-dimensional space is not metrizable
general-topology
functional-analysis
weak-convergence
Coequalizers are quotient maps
general-topology
category-theory
The subset $[a, b)$ of $\Bbb R$ is neither open nor closed.
general-topology
Help Interpreting a Problem Concerning the Union of Connected Sets
general-topology
elementary-set-theory
Two tori $\mathbb C/L$ and $\mathbb C/L'$ are isomorph if $L=L'$
abstract-algebra
general-topology
complex-analysis
riemann-surfaces
lower limit topology to the metric topology
general-topology
Is this space $T_1$ and satisfies property $P$?.
general-topology
Proving that $\mathbb{Q}$ is neither open or closed in $\mathbb{R}$
real-analysis
general-topology
Hausdorff space if no net converges to two different values
general-topology
limits
separation-axioms
nets
Why is empty set an open set?
general-topology
Filters and surjectivity of functions
general-topology
filters
Nerve Theorem: Is the finite union of closed convex sets triangulable?
general-topology
convex-analysis
computational-mathematics
Topologies of Matrix Groups & Algebras
linear-algebra
general-topology
matrices
algebraic-topology
lie-groups
Why is unwinding a string not a continuous transformation?
general-topology
Subfunctor of a hom functor in a Boolean topos
general-topology
category-theory
boolean-algebra
sheaf-theory
topos-theory
Prove that Voronoi cells are path-connected
general-topology
geometry
functions
computational-geometry
Completion with respect to stronger norm is no subset?
general-topology
functional-analysis
metric-spaces
normed-spaces
Example of infinite , compact, path connected metric space $X$ which is homeomorphic to $X \times X$?
general-topology
metric-spaces
compactness
path-connected
Closure as a map
general-topology
How to use a continuous function on $[0,1]$ to prove another set is connected?
general-topology
connectedness
path-connected
Continuity of a function from a one-point compactification to a cone
general-topology
geometry
compactness
geometric-topology
quotient-spaces
Prove two metrics are equivalent?
general-topology
Product Topology/Continuity Confusion
general-topology
continuity
Quotient space and quotient map
general-topology
metric-spaces
compactness
quotient-spaces
topology generated
general-topology
Is every open set a union of (any number of) sets that are homeomorphic to the entire space?
general-topology
metric-spaces
To get a amall strip near the boundary
general-topology
functional-analysis
analysis
multivariable-calculus
Decomposing a topological space into closed subspaces
general-topology
category-theory
(Topology) In the real line $R^1$ a set is open if and only if it is a union of open segments (a,b).
general-topology
Imagining inside of complement set
general-topology
elementary-set-theory
Find the boundary and interior of $C= A \cup B$?
general-topology
Soft question - a subset of a Hilbert space endowed with subspace topology
general-topology
functional-analysis
soft-question
weak-topology
"Inherits as a Subspace" in Toplogy
general-topology
Uniqueness in Baire property representation for compact Hausdorff spaces
general-topology
descriptive-set-theory
baire-category
Is below set is open In $\mathbb{R^2}$
general-topology
Induced Borel $\sigma$-algebra.
general-topology
measure-theory
borel-sets
Show that a function is continuous in one variable.
real-analysis
general-topology
multivariable-calculus
continuity
Prob. 9, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: If $X$ and $Y$ are connected and if $A$ and $B$ are proper subsets of $X$ and $Y$, resp., then
general-topology
proof-verification
connectedness
Find $f : [0,1] \to \mathbb R$ s.t. $\{x \in [0,1]$, for all neighborhood of $x$, there exist $y$ and $z$ in it s.t. $f(y)f(z)<0 \}\ $ is uncountable
real-analysis
general-topology
continuity
Find a topological space $X$ which is connected but has three path components.
general-topology
connectedness
path-connected
A question regarding complements of dense subsets.
general-topology
metric-spaces
Let $\mathbb{R}$ with cofinite topology then infinite subset of $\mathbb{R}$ is?
general-topology
Show that $(\Bbb R^2,T_1)$ and $(\Bbb R^2,T_2)$ are not homeomorphic
general-topology
Why Does the Definition of a Topology via Neighborhoods Include This Axiom
general-topology
definition
Is $X$ a $T_1$ space? (i.e., given a pair of distinct points in $X$, does each one of them have a neighborhood not containing the other?)
general-topology
Can someone help me understand this proof of a modified version of Brouwer's theorem?
real-analysis
general-topology
functions
Unit ball in $l^{2}$ is bounded and closed but not compact.
general-topology
functional-analysis
metric-spaces
lp-spaces
Brouwer theorem (specific version)
general-topology
fixed-point-theorems
fixedpoints
Definition of 'saturated' set?
general-topology
Associated Bundles: Necessity of Action on Left and Right?
general-topology
differential-geometry
fiber-bundles
Possible groups for fundamental group of $\mathbb{R}$
general-topology
fundamental-groups
Homeomorphism of $K$ and $K\cup \{0\}$
general-topology
proof-verification
every open set in the extended real line ($\overline{\mathbb R}$) is a countable union of segments
real-analysis
general-topology
proof-verification
Prob. 8, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: Is $\mathbb{R}^\omega$ connected in the uniform topology?
general-topology
connectedness
Prob. 2, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: If $J$ is a well-ordered set and if $\left\{ X_\alpha \right\}_{\alpha \in J}$ . . .
general-topology
connectedness
Difference between topology and sigma-algebra axioms.
general-topology
measure-theory
soft-question
To introduce a metric on $\mathbb{R}$, such that $f(x) = x^2$ is uniformly continuous
general-topology
metric-spaces
uniform-continuity
What is the relationship between topological gluing and quotient space?
general-topology
Show that $\mathcal{A}$ is dense in $C([0,2016])$.
real-analysis
general-topology
metric-spaces
Example of a non-compact manifold
general-topology
geometric-topology
fixedpoints
Simple question on continuos function & Hausdorff space
general-topology
Which of the following sets are necessarily perfect, connected?
real-analysis
general-topology
proof-verification
compactness
Is the complement of a bounded set bounded?
real-analysis
general-topology
metric-spaces
Necessary/sufficient condition for directional derivative to be zero (infinite-dimensional optimization)
general-topology
convex-analysis
convex-optimization
Given a metrizable topological space $(X,\tau)$, how to construct a metric that induces $\tau$?
general-topology
metric-spaces
Proof verification: topology on the extended real line
real-analysis
general-topology
proof-verification
from weakly sequentielly continuous non linear map to continuous map
general-topology
functional-analysis
weak-topology
Characterisation of Borel hierarchy.
general-topology
descriptive-set-theory
borel-sets
some confusion in symbolic general topology
general-topology
Homeomorphic function spaces arising from homeomorphisms between domains and images?
general-topology
Connected set in R^2
general-topology
connectedness
Strongly separating hyperplane
linear-algebra
general-topology
Is $\overline{A}$ path-connected?
general-topology
Why a metric tensor is considered a tensor as opposed to a matrix?
general-topology
metric-spaces
tensors
Given an element $ \ (a,b) \in A \ $ describe its equivalence class $ \ [(a,b)] \ $ geometrically
general-topology
elementary-set-theory
Difference between basis and subbasis of a topology
general-topology
Show that the addition function for two elements in $\mathbb{R}^\omega$ is continuous
general-topology
continuity
product-space
topological properties of sets $\textit{Iso}(X,d)$ , $\textit{Iso}_{\epsilon} (X,d)$
general-topology
metric-spaces
isometry
Do 2 sets exist that are neither open nor closed, but their union is both open and closed.
general-topology