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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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Is every open interval a union of half open intervals?

I am reading lower limit topology on Wikipedia, which states that the lower limit topology [...] is the topology generated by the basis of all half-open intervals $[a,b)$, where a and b are real numbers. [...] The lower limit topology is finer…
Geoffrey Critzer
  • 3,711
13
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2 answers

Closure of a subset of a subspace of a topological space

Let $X$ be a topological space. Let $Y$ be a subspace of $X$. Let $T$ be a subset of $Y$. Let $\overline T$ be the closure of $T$ in $X$. Then $Y \cap \overline T$ is the closure of $T$ in $Y$?.
Makoto Kato
  • 42,602
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Set of limit points of a subset of a Hausdorff space is closed.

Let $X$ be a Hausdorff space and $A\subset X$. Define $A'=\{x\in X\mid x\text{ is a limit point of }A\}$. Prove that $A'$ is closed in $X$. Relevant information: (1.) Every neighborhood of a point $x\in A'$ contains a point $y\in A'$ distinct…
John J
  • 131
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2 answers

In which spaces can one "flip" the topology?

This is just out of curiosity: In which topological spaces can I relabel all closed sets as open and all open sets as closed and still obtain a valid topology? For example, both the discrete and the indiscrete topolgy can be "flipped" for finite…
HRSE
  • 333
13
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2 answers

Partitioning $\mathbb{R}^2$ into disjoint path-connected dense subsets

Does there exist a partition of the plane into $n=3$ (or more generally $n\ge 3$) disjoint path-connected dense subsets? Note that the answer is yes if "path-connected" is replaced by "connected", as shown here. The linked question also shows that…
Lukas Geyer
  • 18,259
13
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1 answer

If a set is open, its "slices" are also open. Converse?

Suppose $X$ and $Y$ are topological spaces and $U \subseteq X \times Y$. Now, I managed to prove that whenever $U$ is open all "slices" along $Y$ are also open: \begin{equation} U \; \text{is open} \implies \forall \, x \in X : \left \{ y \in Y :…
iolo
  • 1,167
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2 answers

Conditions on a topological space implying that it has a minimal basis

This is a question I've had in the back of my mind for a while, motivated by curiosity. Let $(X,\tau)$ be a topological space, and consider $\mathfrak{F}=\{B\subseteq\mathcal{P}(X):B \text{ is a basis for }\tau\}$ partially ordered by inclusion.…
Alessandro Codenotti
  • 12,285
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2 answers

There is no difference between a metrizable space and a metric space (proof included).

Willard says, "Whenever $(X,\tau)$ is a topological space whose topology $\tau$ is the metric topology $\tau_{\rho}$ for some metric $\rho$ on $X$, we call $(X,\tau)$ a metrizable topological space." I think giving a proof is the best way to…
user41728
  • 1,560
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1 answer

How do the closed subsets in the product topology look like

I know that the open subsets in the product topology of $X=X_1\times X_2\times...\times X_3$, where $X_1,X_2,...,X_n$ topological spaces, are the union of subsets of X: $U_1\times U_2\times ...\times U_n$, where $U_1,U_2,...,U_n$ are open subsets of…
user42912
  • 23,582
13
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2 answers

Show that the Topologies of $\mathbb{R}_l$ and $\mathbb{R}_K$ are not comparable.

Here, $\mathbb{R}_l$ is the lower limit topology on $\mathbb{R}$ and $\mathbb{R}_K$ is the K-topology on $\mathbb{R}$. I understand the proof that these topologies are strictly finer than $\mathbb{R}$, but I am at a loss to begin how to show they…
madisonfly
  • 776
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2 answers

Every point closed $\stackrel{?}{\Rightarrow}$ space is Hausdorff

If a topological space is Hausdorff, then every point is closed. Is the converse true? Edited: Let $G$ be a topological group and $H$ the intersection of all neighborhoods of zero. Since every coset of $H$ is closed, every point of $G/H$ will be…
Manos
  • 25,833
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2 answers

Is the set of irrationals separable as a subspace of the real line?

I am trying to find an example of a separable Hausdorff space which has a non-separable subspace. This led me to ask the question in the title: is the set of irrationals, regarded as a subspace of the real line, separable or non-separable? A space…
Alex Petzke
  • 8,763
13
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1 answer

Product of quotient map a quotient map when domain is compact Hausdorff?

Suppose that $X$ is a compact Hausdorff space and that $q : X \to Y$ is a quotient map. Is it true that the product map $q \times q : X \times X \to Y \times Y$ is also a quotient map? Note I did not assume that the quotient space $Y$ was Hausdorff…
Mike F
  • 22,196
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1 answer

Is $C[0,1]$ normal with the topology of pointwise convergence?

I was asked a question by someone the other day regarding the topology of pointwise convergence, and I can't seem to get anywhere with it. I was wondering if anyone could be of any assistance... The question was: is $C[0,1]$, the set of continuous…
Tom H
  • 664
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1 answer

Munkres, Chapter 2, question on locally finite family of sets

I've been working through the Munkres Topology text on my own, and I am not sure if the following argument is correct. Fishing around the internet a bit for some alternative answers and it looks like the approaches others have taken to this problem…
mrmingus
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