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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
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Equivalence relations on the space $X$

Suppose $X = \left(\mathbb{R} \times \{0\}\right) \cup \left(\mathbb{R} \times \{ 1 \}\right) $ We define $$(x, 0) \sim \left( \frac1x, 1 \right),\ \forall x \ne 0 $$ So, the question is, what space do we get under this equivalence relation? I'm…
ILoveMath
  • 10,694
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Why does a constructible set in a Noetherian topological space contain an open subset dense in its closure?

In a Noetherian topological space, a constructible set is a finite union of locally closed sets. This is a conclusion on constructible sets: Every constructible set contains a dense open subset of its closure. Now neglect the Noetherian condition,…
ShinyaSakai
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Construction of Lakes of Wada

At each step of the construction of Lakes of Wada we extend a lake (an open set in the open unit square) so that no point of the land (the complement of all the lakes) is farther than a given small positive number (depending on the step) from the…
curious
  • 309
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2 answers

Local Homeomorphism of the $S^2$ sphere to $R^2$

I try solving the following excercise: Show by stereoscopic projection that the $S^2$ sphere is locally homeomorphic to $R^2$. I tried to solve this by using the cotangens function on the two angles defining the surface of the ball by the…
Helium
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Prove that this is a topology

I have to prove that the following family defined in $\mathbb{R}^2$ is a topology. $\tau= \{U\subseteq \mathbb{R}^2:$ for any $(a,b) \in U$ exists $\epsilon >0 $ where $[a,a+\epsilon] \times [b-\epsilon, b+\epsilon]\subseteq U\}$ I have started in…
Blanca
  • 741
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A function which is continuous at everywhere in its domain, but diff only at one point

Suppose a real valued function $f:\mathbb R \to \mathbb R$ is continuous everywhere. Is it possible to construct $f$ that is differentiable at only one point? If possible give an example also.
asha
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Showing two spaces homeomorphic

Let us consider two subspaces of $(\mathbb R^2, \tau)$: $A=\{(x,y)\in \mathbb R^2: 0
Anupam
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2 answers

Construct a topological embedding

For an arbitrary discrete space X, construct a compact topological space Y and a topological embedding Y. I am think about construct $X={0,1}$ equipped with the discrete topology. Topology on Y={∅,{0,1},{1}} which is the Sierpinski space. Thus Y is…
manzujizhe
  • 175
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2 answers

Why, precisely, is $\{r \in \mathbb{Q}: - \sqrt{2} < r < \sqrt{2}\}$ clopen in $\mathbb{Q}$?

Just going over some old notes and I realized I always took this for granted without actually fleshing out exactly why it is true. The set of all r is closed in $\mathbb{Q}$, because the set of all r is just all of the rationals in that interval,…
r123454321
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Uncountable subsets of compact and separable metric spaces?

Is it true that an uncountable subset of any compact metric space condenses at uncountably many points of itself? How about of any separable metric space? I know that the first fact is true when you take the metric space $\mathbb{R}$, but is it true…
r123454321
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3 answers

Showing that a function is not a homeomorphism

Consider the function $f: [0,1) \rightarrow \mathbb{C}$ given by $f(t) = e^{2\pi i t}$. I must show that the function $f^*: [0,1) \rightarrow \mathrm{im}(f)$ is not a homeomorphism, given the standard topologies on both sets, but I am not sure how…
Bachmaninoff
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For which intervals $[a,b]$ in $\mathbb{R}$ is the intersection $[a,b]\cap Q$ a clopen subset of the metric space $\mathbb{Q}$?

For which intervals $[a,b]$ in $\mathbb{R}$ is the intersection $[a,b]\cap Q$ a clopen subset of the metric space $\mathbb{Q}$? My answer is : $[a,b]\cap\mathbb{Q}$ is a clopen subset iff $a,b \in (\mathbb{R}\backslash \mathbb{Q})$, since if $a,b…
user98727
  • 79
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2 answers

Every compact subspace of a Hausdorff space is closed

I'm studying the Topology-Munkres book and there is a Theorem that states that Every compact subspace of a Hausdorff space is closed and I was wondering if there is any example where the "Hausdorff" condition is not needed, I mean could you give an…
user89935
  • 61
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Boundary of union of two sets equals the union of their boundaries

Plesae give some hint to solve the following problem: If $A$ and $B$ are two subsets of a topological space $X$ such that $\overline{A}\cap \overline{B}=\emptyset$, then $\partial(A\cup B)=\partial A\cup \partial B$, where $\partial A$ denotes the…
Anupam
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2 answers

Prove: A subset V of $\mathbb{R}$ is open iff V is equal to a union of open intervals

The proof of the theorem is given to me in the book but I need some clarification about specific aspects of the proof that the book thinks is trivial: $\Rightarrow$ Assume V is a open set of $\mathbb{R}$ If V is the empty set then V is trivially…
Gamecocks99
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