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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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$\Bbb R^{\omega}$ in the box topology is not metrizable

Munkres Topology section 21 example 1 shows that $\Bbb R^{\omega}$ in the box topology is not metrizable. (The hidden problems have been solved, but I have another new problem. Please see below.) I don't understand this sentence "the point $a_n$…
user546106
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How can we show that the dyadic rationals are dense in $\mathbb R$?

Numbers of the form $ \dfrac{m}{2^{n}} $, where $ m $ is an integer and $ n $ is a non-negative integer, are called dyadic rational numbers. How can one show that the dyadic rationals are dense in $ \mathbb{R} $?
ccc
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Cantor Set (Hausdorff)

Is the Cantor set Hausdorff? Which separability axioms it has? I know that $\mathbb R$ are Hausdorrf is it enough? Like the Cantor set is compact if it was Hausdorff then the Cantor set would be normal and regular, and it would be T1 for inherited?…
Kc2
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Do homeomorphisms preserve closures of subspaces?

Let $X$ and $Y$ be topological spaces and let $A \subseteq X$ be a subspace of $X$. Suppose $A$ is homeomorphic to some subspace $B \subseteq Y$ of $Y$. Let $f$ explicitly denote this homeomorphism. If $f : A \to B$ is a homeomorphism, does $f$…
Perturbative
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A space whose powers "generate" all spaces via quotients

I've been asked to prove (or disprove) that there exists a topological space $X$ with the following property: For every space $T$ there is a set $I$ and a homeomorphism $T\cong A$ where $A$ is a quotient of a subspace of $X^I = \prod_{i\in I} X$…
fosco
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Homeomorphism between homeomorphic spaces

I have a practice exam problem that asks to prove or disprove the following statement. Every continuous bijection between homeomorphic spaces is a homeomorphism. Now based on everything I know about topology, I feel like I have several ways to…
ai.jennetta
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Maximum (or upper bound on the) weight of a topology on a set X.

Let $X$ be a set and $\tau$ a topology on $X$. Let's define the weight of $\tau$, $w(\tau)$, as the minimum cardinality of a basis for $\tau$. 1) What is the supremum of the weights of all topologies on $X$? 2) Is this supremum a maximum? 3) Is…
Anguepa
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What is the point of a local base?

What's all the fuss about local bases? I find that everywhere I look people are confused of the notion of a local base, and frankly I am as well, because it seems to me it's equivalent an incredibly simple formulation, but everyone else expounds…
user3002473
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What's so discrete about discrete topology?

I am a beginner at topology. I recently learned about discrete topology. But the definition of discrete topology doesn't convey anything 'discrete' to me. Is it just whimsically named like top, bottom quarks in physics or does discreteness has…
Partha Sarker
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Closed sets in the lower limit topology.

Would an interval of the form $[a,b]$ be closed in the lower limit topology $\mathbb{R}_\ell$. Here is why I think it is: Because $\mathbb{R}_\ell$ is finer than the standard topology on $\mathbb{R}$, then all the basis elements of this standard…
user193319
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clarification errata in Munkres Topology?

While reading the second edition of Munkres' Topology, I came across this (page 129): Theorem 21.1 Let $f: X \rightarrow Y$; let $X$ and $Y$ be metrizable with metrics $d_X$ and $d_Y$, respectively. Then continuity of $f$ is equivalent to the…
Vien Nguyen
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When does the product topology have a countable base?

Could any one tell me how to prove this one? The product topology has a countable base if and only if the topology of each coordinate space has a countable base and all but a countable number of coordinate spaces are indiscrete
Myshkin
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Continuous bijection from $(a,b) \to S^1$?

This question started bothering me after working on an exercise. I know that there cannot be a contiuous bijection $S^1 \to (a,b)$ because if there was it would be a homeomorphism but $S^1$ and $(a,b)$ are not homeomorphic. But the theorem that…
self-learner
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Prob. 2, Sec. 20, in Munkres' TOPOLOGY, 2nd ed: The dictionary order topology on $\mathbb{R} \times \mathbb{R}$ is metrizable.

Here's Prob. 2, Sec. 20 in the book Topology by James R. Munkres, 2nd edition: Show that $\mathbb{R}\times \mathbb{R}$ in the dictionary order topology is metrizable. The dictionary order on the set $\mathbb{R} \times \mathbb{R}$ is defined as…
Saaqib Mahmood
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Why should a topological space itself be open?

For convenience, let $X$ be our space. Specifically, can anyone name a few desirable properties or theorems that would fail if $X$ weren't required to be open? More generally, is there a part of topology that would completely fall apart? It seems to…
user33661
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