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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.

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Dimensions and Space-filling curves

If a space-filling curve can continuously map a 1-dimensional interval onto a 2-dimensional region, then what actually makes the region 2-dimensional? Doesn't this mean that only 1-dimension is required to actually describe all the…
Grant
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Prove $\mathbb{R}^3$ is not the product of two identical topological spaces

I can only prove this for $\mathbb{R}$: If $\mathbb{R}\cong T\times T$, then $T$ embeds in $\mathbb{R}$ as a closed subspace (e.g. $T\times pt$). Since $\mathbb{R}$ is connected, so is $T$. So $T$ must be a closed interval, then $T\times T$ is a…
Anonymous Coward
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Compact subset of an open set

Let $X$ be complete separable metric space and $A\subset X$ is open. Does it mean that there is a compact subset of $A$? My solution is the following: since $A$ is open there is $B(x,r)\subset A$, then $\overline{B(x,r/2)}\subset A$ where the closed…
Ilya
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Every locally compact, second countable Hausdorff space has a countable basis of open sets with compact closure

Let $X$ be a locally compact, second countable Hausdorff space. I want to prove that it has a countable basis of opens with compact closure, and that this basis can be extracted as a subset of any basis of its topology by restricting to opens with…
ubugnu
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Prove: the set of zeros of a continuous function is closed.

Prove: the set of zeros of a continuous function is closed. And should the function on a closed interval?
HyperGroups
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“Every open ball is closed" and "every closed ball is open", does one imply the other?

In a metric space $(X,d)$, by a closed ball I mean a set of the form $\{y: d(y,x)\le r\}$, where $r > 0$. A common example where every open ball is a closed set and every closed ball is an open set is the ultrametric spaces: the metric spaces where…
Jianing Song
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Is there a purely topological proof that a certain topological space derived from logical compactness is compact?

Let $L$ be a first order language, and let $S_{L}=\{\sigma:\sigma\;\mbox{is an $L$-sentence}\}$. Also, by logical compatness I mean the Compactness Theorem of first order logic. Compactness Theorem: Let $\Sigma$ be a set of $L$-sentences. $\Sigma$…
John
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Questions about open sets in ${\mathbb R}$

Consider the following problem: Let ${\mathbb Q} \subset A\subset {\mathbb R}$, which of the following must be true? A. If $A$ is open, then $A={\mathbb R}$ B. If $A$ is closed, then $A={\mathbb R}$ Since $\overline{\mathbb Q}={\mathbb R}$, one can…
user9464
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Double-cover of a Klein-bottle-esque Space

I'm trying to complete the following exercise I found in a topology book: Construct a space A which is path-connected with fundamental group equal to $\langle r,s | r^2 s^3 r^3 s^2 = 1\rangle$, and find a unique (connected) double-covering B. Find…
Roy M.
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Finite union of compact sets is compact

Let $(X,d)$ be a metric space and $Y_1,\ldots,Y_n \subseteq X$ compact subsets. Then I want to show that $Y:=\bigcup_i Y_i$ is compact only using the definition of a compact set. My attempt: Let $(y_n)$ be a sequence in $Y$. If $\exists 1 \leq i…
user42761
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Do distance-preserving maps from $\mathbb R^2 \rightarrow \mathbb R$ exist?

So, I know that it's 'impossible' to have a perfectly bijective map $F:\mathbb R^2 \rightarrow \mathbb R$, but I was wondering nevertheless: what would the 'best' possible map be, that is closest to being bijective? Additionally, can you make $F$…
Naren
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Meaning of "holes" counted by homology groups

In a lot of more or less informal introductions to simplicial homology often the groups $H_k(X)$ of a topological space or CW space are introduced as groups which "counting $k$-dimensional holes". I know that is of course motivated by rather…
user7391733
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Examples of metric spaces which are not normed linear spaces?

Give an example of a metric space which is not a normed linear space. Justify your example.
Champa Saikia
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Is projection of a closed set $F\subseteq X\times Y$ always closed?

If we have closed subset $F$ of product $X \times Y$ (product topology) does it mean that $p_1(F)$ (projection on first coordinate) is closed in $X$ and $p_2(F)$ in $Y$ are closed? If not, why not (some counterexample or explanation please =)
Meow
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Is the image of a smooth homeomorphism diffeomorphic to the domain?

This was motivated by the definition of a smooth embedding as an injective smooth immersion that is a topological embedding; I see no reason a-priori for the image of an embedding as defined to be diffeomorphic to the original manifold unless this…
nolatos
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