Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
6
votes
2 answers

Are there nice circumstances under which connectedness of interior and boundary imply connectedness?

Munkres problem 24.11: If $A$ is a connected subspace of $X$, does it follow that $\operatorname{Int}A$ and $\operatorname{Bd}A$ are connected? Does the converse hold? I've answered these questions, but I'm wondering if the converse might hold if…
dfeuer
  • 9,069
6
votes
4 answers

$[0, 1)$ and $S^1$ not homeomorphic?

Let $f:[0, 2\pi) \to S^1 = \{(x, y): x^2 + y^2 = 1\}$ be such that $f(t) \to (\cos t, \sin t)$ $f$ is a continuous bijection but it is NOT a homeomorphism. I suppose the only point of contention is $(1, 0)$ in $S^1$. Is it because there is no open…
sonicboom
  • 9,921
  • 13
  • 50
  • 84
6
votes
2 answers

boundary of a connected component

Let $X$ be a topological space, and $G$ an open subset. If $E$ is a connected component of $G$, then is the boundary of $E$ is contained in that of $G$? I know that it is true if $X$ is locally connected. But I suspect the statement is generally…
ksj03
  • 61
6
votes
1 answer

Linear continuum is convex

Definition. A simply ordered set $L$ having more than one element is called a linear continuum if the following hold: (1) $L$ has the least upper bound property (2) If $x < y$, there exists $z$ such that $x < z < y$. A subspace $Y$ of $L$ is said to…
nomadicmathematician
  • 5,511
6
votes
2 answers

Is a compact Hausdorff space metrizable? Maybe even complete?

Possible Duplicate: A compact Hausdorff space that is not metrizable Is it true that every topological space $X$ that is Hausdorff and compact is also metrizable? Maybe even complete? What is the relationship between the completeness and the…
resu
  • 1,782
6
votes
2 answers

Surjective Function from a Cantor Set

I recently solved an interesting problem on an midterm. Here is one piece of it: Let $K =\prod_{i=1}^\infty \{0,1\}$ in the product topology, and let $s_1,s_2,\ldots$ be a sequence of positive real numbers such that $\sum_{i=1}^\infty s_i = 1$.…
Cardboard Box
  • 4,997
6
votes
1 answer

Precise definition of the boundary set

I am confused about definition of the boundary set. Definition: Given set $A$ and its complement $A^c$; Point $x$ belongs to the boundary of $A$ if every open ball centred at $x$ contains points of $A$ and $A^c$. Now to my confusion: Taking as an…
Leon
  • 475
6
votes
1 answer

$f\colon X \to Y \times Z$ is continuous iff its component functions are

I have a homework question that asks me to prove that a given function $f : X \rightarrow Y \times Z$ is continuous if and only if its component functions $f_Y : X \rightarrow Y$ and $f_Z : X \rightarrow Z$ are continuous. I think I solved it, and…
Paul Accisano
  • 235
6
votes
2 answers

an homeomorphism from the plane to the disc

Someone asked me to give an explicit homeomorphism between $\mathbb C$ and the unit disc. I gave him the following answer: we look at $\mathbb C$ as $\mathbb R^2$. The map $x\mapsto \tan (\pi x/2)$ is an homeo from $(-1,1)$ to $\mathbb R$ which…
palio
  • 11,064
6
votes
3 answers

How do you prove that $\mathsf{bd}(\mathsf{bd}(\mathsf{bd}(W)))= \mathsf{bd}(\mathsf{bd}(W))$

$\newcommand{\bd}{\operatorname{bd}}$Prove that the $\bd(\bd(\bd(W)))=\bd(\bd(W))$ where $W$ is a subset of the topological space $(X,\mathscr{T})$.
joewoo89
  • 61
6
votes
3 answers

Any two disjoint open sets are the interior and exterior of some set

For a topological space $X$, given any two open sets $A,B$, there is a set $S\subseteq X$ such that $\DeclareMathOperator{\ntr}{int}\ntr S=A$ and $\DeclareMathOperator{\ext}{ext}\ext S=B$. Is this true for $X=\Bbb R^2$? If so, what topological…
Mario Carneiro
  • 27,438
6
votes
2 answers

Determine the closure of the set $K=\{\frac{1}{n}\mid n\in\mathbb N\}$ under each of topologies

The questions are the following: Consider the five topologies on the real line $\mathbb R$: $\mathcal T_1$: the standard topology $\mathcal T_2$: the $K$-topology $\mathcal T_3$: the finite complement topology $\mathcal T_4$: the upper limit…
Jihyun Kim
  • 219
6
votes
1 answer

Constructing paths which are not constant on any interval

Suppose $X$ is a Hausdorff topological space (or metric space if you like) and $f:[0,1]\to X$ is any non-constant path. It could be that $f$ is constant on a closed interval $[a,b]$, and it is possible to collapse this interval to a point in $[0,1]$…
J.K.T.
  • 1,536
6
votes
0 answers

How to show that the Tychonoff product is associative?

Let $\{ X_t : t \in T\}$, be a family of topological spaces. Suppose thst $T = \bigcup \{ T_s : s \in S \}$, where $T_s \neq \emptyset $ for all $s \in S$, and $T_s \cap T_{s'} = \emptyset$ if $s \neq s'$. How can I prove that $\Pi_{t \in T} X_t$…
topsi
  • 4,222
6
votes
1 answer

Definition of a Manifold with a boundary

Lee in his book on topological manifolds says that An $n$-manifold with a boundary is a second countable Hausdorff space in which any point has a neighborhood which is homeomorphic either to an open subset of $\mathbb R^n$ or to an open subset of…
SBF
  • 36,041
1 2 3
…
99 100