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
5
votes
2 answers

Why is every map to an indiscrete space continuous?

Show that if $Y$ is a topological space, then every map $f:Y \rightarrow X$ is continuous when $X$ has the indiscrete topology. Proof: Assume $X$ has the indiscrete topology, $T=\{\varnothing,X\}$. $f$ is continuous if $f^{-1}(V)$ is an open subset…
user8603
  • 405
5
votes
2 answers

How do I show that $f: [0,1) \to S^1$, $f(t) = (\cos(2\pi t), \sin(2\pi t))$ is not a homeomorphism?

We have a unit circle with subspace topology induced from $\mathbb{R}^2$ How do I show that $f: [0,1) \to S^1$, $f(t) = (\cos(2\pi t), \sin(2\pi t))$ is not a homeomorphism? So we have two topological spaces, $\mathbb{R}$ with the standard…
sarah
  • 1,281
5
votes
1 answer

Disjoint closures in a metric space implies the space is discrete

Let $X$ be a metric space such that for every pair of disjoint open sets $U$ and $W$ we have $\overline{U} \cap \overline{W} = \emptyset$. How to prove $X$ is a discrete space?
user10
  • 5,688
5
votes
3 answers

Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$

Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$? Does the converse hold? I start by trying to understand what $\mathbf{Bd}$ means in "formal terms" (i.e. $\bar{A} \cap…
Siyanda
  • 2,549
  • 10
  • 35
  • 49
5
votes
2 answers

Quotient space of S1 is homeomorphic to S1

$S^1=\{z\in\mathbb{C}\mid |z|=1\}$, let $w\sim z$ iff $w=z\vee w=-z$ (identifying antipodal points). Prove $S^1/\sim$ is homeomorphic to $S^1$. Which function should be used to construct a homeomorphism? I am not good at analysis. Thanks!
Kaa1el
  • 2,058
5
votes
0 answers

Topologist Sine Curve

I am trying to prove that the topologist sine curve is not path connected. I think I have a proof but my proof relies on Intermediate Value Theorem. So, I was wondering if there is a way to prove it without using Intermediate Value Theorem. Thank…
user20694
  • 165
5
votes
3 answers

Show that if $U$ is a nonempty open set in $\mathbb{R}$, then $\overline{U}$ = $\mathbb{R}$.

Let $\mathbb{R}$ be a topological space with topology $T = \{ U \subseteq \mathbb{R} : U^c \ \text{is finite} \} \bigcup \{\emptyset \}$. Show that if $U$ is a nonempty open set in $\mathbb{R}$, then $\overline{U} = \mathbb{R}$. So I suppose…
sarah
  • 1,281
5
votes
1 answer

$T_2$ spaces and isolated points

Is there a topological Hausdorff space with an infinite number of isolated points such that any infinite set of isolated points have an infinite number of limit points !? (Of course it would be impossible if a limit point is a limit of a sequence.)
user41304
  • 613
5
votes
3 answers

equivalent definitions of closure

This is a topology question. First of all, I am sorry this is a really dummy question. As a math student, it is a shame I haven't taken any courses in topology. A closure of a set $A$ is usually defined as $\operatorname{int}(A)\cup \partial A $. I…
newbie
  • 3,441
5
votes
2 answers

Are these bases for a topology?

I have the following topology : $$\tau= \Bigl\{U\subseteq \mathbb{R}^2: (\forall(a,b) \in U) (\exists \epsilon >0) \bigl([a,a+\epsilon] \times [b-\epsilon, b+\epsilon]\subseteq U\bigr)\Bigr\}$$ Are these a basis for the previous topology: $\beta_1=…
Blanca
  • 741
5
votes
4 answers

Prove that $[0,1]$ is not a compact subset of $\mathbb{R}$ with the lower limit topology, i.e. open sets are of the form $[a,b)$.

Prove that $[0,1]$ is not a compact subset of $\mathbb{R}$ with the lower limit topology, i.e. open sets are of the form $[a,b)$. My question is will different topology affect compactness of a set? If this is so, why? At first, when I see this…
Idonknow
  • 15,643
5
votes
1 answer

Understanding Counter example for "product of two quotient maps is a quotient map" from Ronald Brown

One can prove that the product of two open quotient maps is a quotient map. Ronald Brown gives a counter example for the fact that this is in general not true for arbitrary quotient maps, in his book Topology and Groupoids on page 111. The counter…
user102383
  • 53
5
votes
1 answer

Any Subset of a Set Containing No Accumulation Point is Closed

Let $(X,\tau)$ be a topological space. Suppose that $B\subseteq X$ has no accumulation point. That is, for any $x\in X$ there exists some $U\subseteq X$ such that $x$ is in the interior of $U$ and $B\cap U\cap\{x\}^c$ is empty. Claim: If $C\subseteq…
triple_sec
  • 23,377
5
votes
1 answer

Is compactness of sequentially compact sequential Hausdorff spaces determined by the well ordered space $\omega_1$?

The space $X$ is sequential if for each nonclosed $A \subset X$, there exists a convergent sequence $a_n \rightarrow x$ so that $a_n \in A$ but $x \notin A$. By $\omega_1$ we mean an uncountable well ordered set (with the order topology) so that…
Paul Fabel
  • 629
5
votes
3 answers

Connectedness problem and countability of compact subsets of the Sorgenfrey line

1) Stuck with this problem, can you please help? Show that if $X$ is connected and for a connected subspace $A$ of $X$ we have $X \setminus A = U \cup V$ where $U,V$ are open in $X \setminus A$ and disjoint, then the sets $A \cup U$ and $A \cup V$…
user10
  • 5,688
1 2 3
…
99 100