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
4
votes
2 answers

Homeomorphism example

I'm reading some notes about topology and homeomorphisms and there is an example of a homeomorphism from the unit ball in $\mathbb R^n$ to $\mathbb R^n$. The map $x \mapsto {x \over 1 - |x|^2}$. I assume absolute value means Euclidean norm here.…
newb
  • 1,593
4
votes
1 answer

Prove the set of reals with the Zariski topology is not metrizable.

I cannot use facts involving Hausdorff spaces, as this problem expects knowledge a little bit more elementary. I am mostly confused with the statement "A topological space (X, $\tau$) is metrizable if there exists a metric d such that $\tau$ is the…
1239asd
  • 77
4
votes
1 answer

Trivial Topology and Basis

I recently started studying topology (Royden and Fitzpatrick's Real Analysis - 4th edition) and I stumbled upon this exercise: "Show that the trivial (indiscrete) topology on a set $\, X \,$ has a unique base." Consider a topological space $(X,…
Giordano Ribeiro
  • 339
4
votes
1 answer

Continuous Functions and dense set

Let $X$ be a topological space and $D\subset X$. If for every $f,g\in C(X)$ we have $f(x)=g(x)$ for all $x\in D$ implies $f=g$. Then $D$ is a dense subset of $X$.
shane
  • 381
4
votes
1 answer

Is closure of intersection open in intersection of closures for open subsets?

Let $X$ be a topological space, and $A, B \subset X$ be open in $X$. Is $\overline{A \cap B}$ open in $\overline{A} \cap \overline{B}$? Background: It can be shown that $\overline{A \cap B} \subset \overline{A} \cap \overline{B}$. In general, these…
kaba
  • 2,035
4
votes
1 answer

Homeomorphism from the suspension of $S^{n-1}$ to $S^n$

Let $S(X) = (X \times [-1,1])/R$ denote the suspension of $X$, where the classes of $R$ are $X \times \{1\}, X \times \{-1\}$ and all singletons $\{a\}$, where $a \in X \times (-1,1)$. Show that $S(S^{n-1})$ is homeomorphic to $S^n$. (Don’t confuse…
Wondera
  • 547
4
votes
3 answers

A quick question about Theorem 24.1 of Munkres.

Can someone please help me understand why the underlined statement must be true. Why must there be a $d$ in $B_0$ less than $c$? Why can't $c$ be the smallest element of $B_0$?
Donald Obama
  • 131
4
votes
1 answer

$f(x)=\begin{cases} 0 & \text{ if } x\in (-\infty,0) \\ 1 & \text{ if } x\in [0,+\infty] \end{cases}$, continuous?

Question: is $f: (\mathbb{R},τ) \rightarrow (Y,σ)$ with $f(x)=\begin{cases} 0 & \text{ if } x\in (-\infty,0) \\ 1 & \text{ if } x\in [0,+\infty] \end{cases}$, continuous? Where $τ$ is the usual topology of $\mathbb{R}$, $Y=\{0,1,2\}$ and…
領域展開
  • 2,139
4
votes
1 answer

Use of choice function in Urysohn's lemma?

Let $A = \mathbb Q \cap [0,1]$. In the proof of Urysohn's lemma, we construct a family of open set such that: $\{V_q\}_{q \in A}$, where if $r_1 < r_2$, then $V_{r_1} \subset \subset V_{r_2}$. The construction was made possible by some choice…
James C
  • 1,338
4
votes
2 answers

Removing countable hyperplanes from a ball in $\mathbb{R}^n$

I would like to claim that an open ball in $\mathbb{R}^n$ cannot be covered by a countable collection of $(n-1)$-dimensional hyperplanes, that in fact excluding the hyperplanes from the ball still leaves a set that contains an uncountable number of…
Joseph O'Rourke
  • 30,327
4
votes
3 answers

Homeomorphic Spaces in Topology

Let $X$ and $Y$ be two topological spaces. Let $f : X \to Y$ and $g: Y \to X$, such that $f$ and $g$ are surjective; $f$ and $g$ are continous. Does this imply that $X$ and $Y$ are homeomorphic? It seems similar to Bernstein's theorem in set…
Kr Dpk
  • 483
4
votes
1 answer

Must $X$ be lindelöf if it has countable network?

A network is like a base, except that its members need not be open sets. A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exists an $M \in \mathcal N$…
Paul
  • 20,553
4
votes
1 answer

Show torus homeomorphic to $S^1 \times S^1$

Prove that $\mathbb{T}^2$ is homeomorphic to $S^1 \times S^1$. Here is my attempt on this. Is this correct? Let $C^*$ be the torus as an identification space of $[0,1] \times [0,1]$ under the quotient map $g:[0,1] \times [0,1] \rightarrow C^{*}$.…
ernesto
  • 549
4
votes
2 answers

Closed sets in a cofinite topology

I'm reading a set theory text book and I've come upon the section on topologies. (An Introduction to Set Theory and Topology by Ronald Freiwald, p. 105). The author defines the cofinite topology in the usual way: T = {O ⊆ X : O = ∅ or X - O is…
tupben
  • 163
4
votes
2 answers

Metric that induces topology distinct from product topology

Find a metric on $\mathbb{R}^n$ that does not induce the same topology as the product topology.My effort: Consider the metric $$d(\textbf{u},\textbf{w})=\begin{cases} 1 &\text{ if } p \neq q\\ 0 &\text{ if } p=q\end{cases}$$ and the open ball…
ernesto
  • 549
1 2 3
…
99 100