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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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On the existence of a nontrivial connected subspace of $\mathbb{R}^2$

In the nontrivial sense, does there exist a connected subspace of $\mathbb{R}^2$ which is a union of a non-empty countable collection of closed and pairwise disjoint line segments each of unit length, i.e. length $1$? What are some good examples, if…
Libertron
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Partitions of $\mathbb{R}^2$ into disjoint, connected, dense subsets.

Does there exist pairwise disjoint, connected, dense subsets $U_1,\dots, U_n \subset \mathbb{R}^2$ such that $U_1\cup \cdots \cup U_n =\mathbb{R}^2$? If $n=1$, then we can take $U_1 = \mathbb{R}^2$. If $n=2$, then we can take $$U_1=…
Adam Lowrance
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Topology on a space starting from topologies on subspaces

I have a question about constructing a topology on a space $X$ starting from topologies defined on a family of subspaces $(X_i)_{i\in I}$ of $X$. Assume that $X$ is a set and $(X_i)_{i\in I}$ is a collection of subsets of $X$. Assume also that for…
Rajai Nasser
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Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed

Consider $\mathbb{R}^2$ as $\mathbb{R} \times \mathbb{R}$ with the product topology. I am simply trying to show that the two projections $p_1$ and $p_2$ onto the first and second factor space respectively are not closed mappings. It seems like…
Alex Petzke
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When is uniform space normal

We know that metric spaces are normal. We also know that a uniform space is Hausdorff if the intersection of all entourages is the diagonal, in which case it is even regular. However, is there a necessary and sufficient criterion to ensure that a…
Ran Wang
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every single point is closed?

If we have a topological space $X$, consider the following: Let $p \in X$. Then $\{p\}$ doesn't have any limit points because it's a single subset of $X$, so $\{p\}$ is closed. Is this correct? So every single point subset in a given topological…
user42912
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$\mathbb{R}^\mathbb{R}$ is not normal

Does anyone know how to prove that $\mathbb{R}^\mathbb{R}$ (with the product topology) does not fulfill the $T_4$ axiom? It would be sufficient to have an uncountable subset $A \subseteq \mathbb{R}^\mathbb{R}$ which is closed and discrete as…
Dune
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Are two spaces manifolds if their product is a manifold?

Possible Duplicate: Decomposition of a manifold For topological spaces $X,Y$, if their product space $X \times Y$ is a manifold, is it necessarily that $X,Y$ are manifolds?
Summer
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How to show a function can or cannot be extended to a compactification?

This comes from Munkres 38.2. Let $Y$ be the compactification of $(0,1)$ induced by $h(x) = (x,\sin(1/x))$. Show that $g(x) = \cos(1/x)$ cannot be extended to this compactification $Y$. Also I wonder in general how to show a function can or cannot…
snsunx
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Is every locally compact Hausdorff space paracompact?

It seems likely that for any open cover, we can construct a locally finite refinement using the local compactness of the space. I can't figure out how to work the construction though, and I'm not yet convinced that there is no counterexample.
notredame523
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Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic?

Suppose that $A,B$ are subsets of $\Bbb{R}^2$. If $A$ and $B$ are homeomorphic and $A$ and $B$ are compact and connected, are their complements homeomorphic?
Jalil
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Are these two definitions of basis equivalent?

Lecture note definition Let $(X, \mathcal{T})$ be topological space, A $basis$ of $\mathcal{T}$ is a collection $\mathcal{B}$ of open sets satisfying the following: For each open set $U$ and for each element $x \in U$, there exists a set $\beta$…
ElleryL
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Can we fit uncountably many nonempty open sets in $\mathbb{R}^n$ such that each point is contained in at most finitely many of them?

This simple question came to my mind the other day: Question: Can we fit uncountably many nonempty open sets in $\mathbb{R}^n$ such that each point of $\mathbb{R}^n$ is contained in at most finitely many of the sets? I spent hours trying to find a…
Patrick
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Are all finite sets closed?

say $X=\{a,b\}$ be a set. The following is a topology on $X$. $\tau=\{\{ a\}, \{a,b\}, \varnothing\}$ Then $b$ is a limit point of $a$, as all open sets $(\{a,b\})$ intersect $\{a\}$ at points other than $b$. Then how is it that all finite sets are…
user3001408
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Having difficulty understanding topological groups.

Let $G$ be a group and $x,y\in G$. We say that a topology $\mathcal{T}$ is a group topology if the functions $$f: G\times G \rightarrow G,\quad (x,y)\mapsto xy$$ and $$g: G\to G,\quad x\mapsto x^{-1}$$ are continuous. We call the pair…
JessicaK
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