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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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Topology for convergent sequences

Let $(X,\tau)$ be a topological space, and consider the family $\mathcal{F}$ of the topologies over $X$ such that the convergent sequences for each $\gamma \in \mathcal{F}$ are the same as the convergent sequences for $\tau$, with the same limits.…
André Caldas
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Understanding the concept of neighborhood basis

The definition of basis and neighborhood basis are: Let $(X,\tau)$ be a topological space, a base of $\tau$ is a subset $\mathfrak{B}$ of $\tau$ such that each open set $A \in \tau$ is union of elements of $\mathfrak{B}$ If $p \in X$, a subset…
some_math_guy
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Boundary of a countable union sets contained in the union of boundaries?

First off, this is related to my previous question. Suppose that I have a countable collection of mutually disjoint open cubes $\{Q_k : k \in \mathbf{Z}^+\}$ in $\mathbf{R}^n$. What can I say about the boundary of the union? That is, what can I say…
Suugaku
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$f: \mathbb{R}^2 \to \mathbb{R}$ a continuous open map, show that for each $x \in \text{range}(f)$, $f^{-1}(x)$ is always uncountable.

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous open map. Show that in fact for each $x \in \text{range}(f)$, $f^{-1}(x)$ is always uncountable. I know that if this was simply a projection onto the one of the two coordinates, this problem…
WayMoreQuestionsThanAnswers
  • 265
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How to prove that $\mathbb R^\omega$ with the box topology is completely regular

How do I prove that $\mathbb R^\omega$ with the box-topology (i.e., the basis are of the form $\prod_n G_n$, where $G_n$ are open in $\mathbb R$) is Completely Regular (i.e. Given a point $a$ and a closed set $F$; one can find a continuous function…
fandango
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Is the union of an arbitrary collection of topological spaces a topological space?

Let me first give a definition. By a separation of a topological space $X$, I mean a pair $U, V$ of disjoint non-empty subset of $X$ whose union is $X$. My question revolves around this well-known theorem of connectedness in topology. Let me quote…
saru
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Closure of a compact set in Regular space $ X $

I am trying to show that closure of a compact set in a Regular space is compact, but I am hitting some hurdles as a compact set in a regular space need not be closed. This is what I started off with: If $ C $ is any compact subset, let $ U $ be any…
Vishesh
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Why is $f:\mathbb{R} \to S^1, f(t)=(\cos(2\pi t), \sin(2\pi t))$ not closed?

I'm studying intro. to topology. I have the following function: we have to topological spaces, $\mathbb{R}$ with the standard topology and $S^1$ with the subspace topology from $\mathbb{R}^2$. I am asked to show that the map $f:\mathbb{R} \to S^1,…
Shay Ben Moshe
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Definition of $\mathbb{R}^\infty$

Question: Why is the topological space $\mathbb{R}^\infty$ defined to be the subset of $\prod_{i=1}^\infty \mathbb{R}_i$ consisting of sequences $(a_i)_{i=1} ^{\infty}$ such at most finitely many $a_i\neq 0$? Why does one insist on the condition…
Holdsworth88
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Topology - Quotient Space and Homeomorphism

Consider the topological spaces $A \subseteq X$. Identify the quotient space $X/A$ as a more familiar topological space and prove its homeomorphic. $X = \mathbb{R}$ and $A = \mathbb{Z}$ My thought was that $X/A$ is homeomorphic to a circle. Is this…
user64013
  • 569
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Connection between connected and compact spaces?

What kind of connection is there between connected and compact subspaces, if any? I am just curious. I know that the image of a compact space under a continuous function is compact and the same holds for connected spaces. But this is not what I am…
elaRosca
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Is Neighborhood an open set?

I am reading a book where it is written that , Let $(X,d)$ be any metric space $a \in X$ then for any $r \gt 0$ the set $S_r(a)$ ={$x \in X$ : $d(x,a) \lt r$} is called an open ball of radius $r$ centered at $a.$ & Let $(X,d)$ be any metric…
Supriyo Banerjee
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Isn't this proof of a theorem about the closedness of a set wrong?

I was reading a proof of the following theorem in my textbook: A set $A$ is closed iff $A' \subseteq A$. Proof: Suppose $A$ is closed and $x \in A'$. If $x \notin A$, then $x\in A^c$, an open set. Thus $\mathcal{N}(x, \delta)\subseteq A^c$ for some…
cppcoder
  • 403
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Equivalent definition of a quotient map

I'm trying to understand why the two definitions of a quotient map are equivalent. Suppose $p:X\to Y$ is a quotient map in the first definition. Then certainly $p$ is continuous and maps all open sets to open sets (in particular it maps saturated…
user557
  • 11,889
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Let $X$ be an infinite set and $τ$ a topology on $X$. If every infinite subset of $X$ is in $τ$, prove that $τ$ is the discrete topology.

I have been thinking about this one for awhile, but I cannot crack it. I think the proof to this is similar to the proof that the topology containing all singleton sets is the discrete topology, in that that proof used the infinite union of…
Matthew Riley
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