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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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Homeomorphism in $\mathbb{R}$ with the upper limit topology.

Consider $\mathbb{R}$ with the upper limit topology (open sets are of the form $(a,b]$) and consider the subsets $(0,1]$ and $(0, +\infty)$ with the corresponding relative topologies. Show that $(0,1]$ and $(0, +\infty)$ are homeomorphic.
Santiago Vargas Rendón
  • 43
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Clarification regarding basis for a topology

This might be super trivial but I if possible would like some clarification on this topic. I am reading from Munkres' Topology, 2nd edition, page 78 (if interested). My question regards what a basis for a topology means. Below is what I am thinking…
user43138
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Proving a continuous function is constant, given some conditions?

So here is my question: I need to prove that a continuous function $f: M \mapsto \mathbb{Z}$, is constant provided that M is connected. I am having trouble understanding this statement; if I set M = $\mathbb{R}$, how is $f$ constant? Am I…
r123454321
  • 2,069
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Closed Subgroups of $\mathbb{R}$

If $F$ is a nonempty closed subset of $\mathbb{R}$ such that $x-y\in F$ for $x,y\in F$, then show that $F=\mathbb{R}$ or $F=\alpha\mathbb{Z}$ for some $\alpha\in \mathbb{R}$.
Anupam
  • 4,908
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Neighborhood and upper limit topology homework help

EDIT: This has been edited so please only take a look at the 2nd question D. I'm not very good with topology but I would like some advice on my homework and also if possible to verify if what I've done is correct. While I'm still very new to…
Nikolaj Kyed
  • 87
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Specific instance of a general selection principle in topology.

I am trying to figure out how to approach a proof of the following claim: For any sequence $\langle U_n: n \in \mathbb{N} \rangle$ of bases for the standard topology on $\mathbb{Q}$, there is a sequence $\langle F_n: n \in \mathbb{N} \rangle$ such…
objectivesea
  • 823
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The $\infty$-dimensional cube, $\infty$-dimensional cross-polytope and their pseudo-boundaries

Let $A = \mathbb{R}^\mathbb{N}$ with the product topology. Let $B = \{ (x_n)_n \in A \mid \forall n : |x_n| \leq 1\} = [-1, 1]^\mathbb{N}$ be the $\infty$-dimensional cube. Let $C = \{ (x_n)_n \in A \mid \sum_{n=1}^\infty |x_n| \leq 1 \}$ be the…
Smiley1000
  • 950
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Is every regular Alexandrov topology a partition topology?

An Alexandrov topology on a set $X$ is a topology in which arbitrary intersections of open sets are open. Equivalently, every point has a smallest open neighborhood. Given a partition on a set $X$, one can form the corresponding partition topology…
PatrickR
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Topology - Is infinite product of quotient maps a quotient map?

Product of two quotient maps need not be a quotient map. However, given two quotient maps $$p : A → B\\ q : C → D$$ if A, B, C, D are locally compact Housdorff then $$p × q : A × C → B × D$$ is a quotient map. Does this still hold for product of…
user63814
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Cover $~E \to B~$ gives a homeomorphism $~E/Aut(E) \to B~$

Let $p : E \rightarrow B$ be a cover s.t. Aut(E) acts transitively on $p^{-1}(b)$ for some fix $b \in B$. Then $E/Aut(E)$ is homeomorphic to $B$ where $Aut(E) \subset Cov(E,E)$. The problem is that I don't even understand how the map $~E/Aut(E) \to…
Down
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Is countable intersection of open sets an open set???

I doing some exercise related to $G_{\delta}$ set and got something confused. From the definition of topology space, finite intersection of finite open sets is an open set. By induction, we can conclude that countable intersection of open sets is…
le duc quang
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Determination of connected set

Prove or disprove: $\lbrace (x,y)\in \mathbb{R}^{2}: x^2+y^3\in \mathbb{R}\setminus \mathbb{Q}\rbrace$ is disconnected with the usual topology in $\mathbb{R}^{2}$.
Anupam
  • 4,908
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Is there any significance or intuition to the fact that the same topology can be produced by different bases with different "shapes"?

In $X = \mathbb R^n$ (finite $n$ to make it simple), we can use either the open balls with the Euclidean metric, $B_d(x, \epsilon)$, or the Box topology (i.e., $n$-dimensional rectangular prisms with finite or infinite sides), as the basis for a…
skymonkey
  • 125
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How can the inverse of the projection mapping in a product topology exist?

I read that the projection mapping $\pi_x: X\times Y\to X$ is continuous. For this function to be continuous, the inverse $\pi^{-1}_x: X\to X\times Y$ has to exist. Take any point $x\in X$. The inverse $\pi^{-1}_x(x)$ has multiple values-…
user67803
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Definition of local homeomorphism

According to Wikipedia : A function $f:X→Y$ between two topological spaces is called a local homeomorphism if every point $x\in X$ has an open neighborhood $U$ whose image $f(U)$ is open in $Y$ and the restriction $f\rvert_U:U→f(U)$ is a…
C2H6
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