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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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Spaces where all singletons are closed

Do spaces where all singletons are closed have a name? I know for example that $\mathbb R$ is one of these spaces since the complement of a singleton $\{x\}$ is $(-\infty,x)\cup (x,\infty)$ which is open. I know also that a space where all…
palio
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Show that two spaces are not homeomorphic

Let $H=[-1,1]\times \{0\}$ and $V=\{0\}\times [-1,0)$ in the plane. Let $T=H \cup V$. Show that $T$ is not homeomorphic to the unit interval $I=[0,1]$. My idea for this problem is that , if we remove a point from the unit interval , we will be left…
the8thone
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embedding of closed 3-manifolds

Prove or disprove that closed 3-manifolds which are not simply connected cannot be embedded in three-dimensional Euclidean spaces. I am not a mathematics major and I am taking introductory topology this semester. But I need to apply this result for…
Srikanth
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Show that the function $f:X\to \mathbb R$ defined by $f(x)=d(x,A)$ is continuous.

Let $d$ be a metric on $X$ and let $A$ be any arbitrary subset of $X$. Show that the function $f:X\to \mathbb R$ defined by $f(x)=d(x,A)$ is continuous. Let $p\in X$. We want to show that for any open set $S$ containing $f(p)$, there exists an…
pxc3110
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Identifying topology

Which of the followings are topology of $\mathbb R$? I believe that 7,8 are while topology, which I think I can understand why. While for 9,10, I believe there're trap, preventing them from being a topology, but I have no clue why are they not.…
JSCB
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Extension of a function of a sphere to the disk.

Let $f:S^{n-1}\rightarrow Y$ be a continuous map from the sphere to a topological space. Why does $f$ have to be nullhomotopic for it to be extendable to the disk? I know this may be a silly question but I don't quite get it.
Sak
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How to show that $X$ has property $S_1(\mathcal O,\mathcal O)$, if and only if $\Sigma_{n=1}^{\infty} X_n$ does.

We say that a space $X$ satisfies $S_1(\mathcal O,\mathcal O)$, if for every sequence of open covers $\{ \mathcal U_n : n \in \mathbb N \}$, there exists a sequence of open sets $\{ U_n : U_n \in \mathcal U_n \text{ for each } n \in \mathbb N \}$,…
topsi
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Subspace topology and open singletons

$(X, T)$ is an infinite topological space, and $C:= ${{$x$}:{$x$}$∈T$}. $C$ is finite. Does $Y:=X\setminus C$ have any open singleton if we consider the subspace topology on $Y$? What if X is Hausdorff?
user136592
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Polygonally connected

Prove that any open connected set in $C[0,1]$ is polygonally connected. (Here $C[0,1]$ is the space of real valued continuous functions on $[0,1]$ with the metric: $d(f,g)=$ $sup${$|f(x)-g(x)| : 0\leq x\leq 1$}; a polygonal path is a path made up of…
WhySee
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Any open set in $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$

I believe the claim is right. But I can only prove for the case where the open set is convex. EDIT: As someone pointed out, it is not true in general. How about "simply connected open set"?
Yan Zhu
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Which sets occur as boundaries of other sets in topological spaces?

Which sets occur as boundaries of other sets in topological spaces? Of course the boundary of a set is closed. But is every closed set in a topological space, the boundary of some set in that space? It is tempting to assert that boundaries have…
N Unnikrishnan
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Is every regular separable space normal?

Is every regular separable space normal? Here are some (standard, I think) definitions: A space is called regular if given any nonempty closed set $F$ and any point $x$ that does not belong to $F$, there exists a neighbourhood $U$ of $x$ and a…
zuriel
  • 603
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A sphere is homeomorphic to a cylinder where the top and bottom bases are identified to single points.

Let $\sim$ denote the equivalence relation on the cylinder $S^1 \times [-1, 1]$ defined by $(v,-1) \sim (v',-1)$ for all $v, v' \in S^1$, and $(v, 1) \sim (v', 1)$ for all $v, v' \in S^1$. Here $S^1$ is the unit circle $\{(x, y) \in \Bbb{R}^2 | x^2…
Artus
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Verifying the Cofinite topology

I'm trying to provee that the cofinite topology is a valid topological space. I've defined it as $$C=\{\emptyset\}\cup\{S\subseteq X: X-S \quad\text{is finite }\}.$$ Now, clearly $\emptyset$ and $X$ are in $C$. I just have to prove that it is…
Millardo Peacecraft
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How to prove that $X$ is not countable implies ,$C_p(X)$ not first countable?

How can I prove, that, Given a Tychonoff topological space $X$, If $X$ is not countable, then, $C_p(X)$ is not first countable? Thank you!
topsi
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