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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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Playing with closure and interior to get the maximum number of sets

Can you find $A \subset \mathbb R^2$ such that $A, \overline{A}, \overset{\circ}{A}, \overset{\circ}{\overline{A}}, \overline{\overset{\circ}{A}}$ are all different? Can we get even more sets be alternating again closure and interior?
mathcounterexamples.net
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Locales with no points

I'm very puzzled by the concept of a locale with no points. I understand that once one switches to the language of open sets and operations on them, points become optional: an open set may or may not have points. More puzzling are locales which…
Tnatsissa H Craeser
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Find the sum of all the number formed by 2,4,6, and 8 without repetition.Number may be of any digit like 2, 24, 684, 4862.

Find the sum of all the number formed by 2,4,6, and 8 without repetition.Number may be of any digit like 2, 24, 684, 4862. My Approach: single digit no formed = 2,4,6,8 sum= 2+4+6+8= 20 two digit= 24,26,28,42,46,48,62,64,68,82,84,86 sum= 660 three…
justin takro
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Simple question on closed sets

A closed set is one which contains all its limit points. Why is $[a, \infty)$ closed? Specifically I don't understand how $\infty$ which is a limit point, but it is not in the set.
bissi
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The space of homeomorphisms of the closed unit interval

Denote by $X$ the space of homeomorphisms of the unit interval $[0,1]$, equipped with the topology of uniform convergence. It is an exercise in a textbook I'm reading to show that $$X \cong \{ 0, 1 \} \times [0,1]^{\omega}.$$ The progress I've made…
Mr. Chip
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When I prove two topologies are same, is it sufficient to prove they have same basis? If so,Why?

$(X, \mathcal{T}_X), (Y,\mathcal{T}_Y)$ be topological space, and let $A, B$ be subset of $X,Y$ respectively. Now, $(X \times Y, \mathcal{T}_{X \times Y})$ be product topology. $A \times B \subset X \times Y$. Consider subspace topology …
ElleryL
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Can this be a way to prove that $\Bbb{R}^2$ and $\Bbb{R}^3$ are not homeomorphic?

The normal way I use to prove that $\Bbb{R}$ and $\Bbb{R}^2$ are not homeomorphic is by removing a point and then using path connectedness. But this method doesn't seem to work for $\Bbb{R}^m$ and $\Bbb{R}^n$ and it ends up that its better if you…
user210387
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Why topology on a set is defined the way it is?

Following is from Wolfram Mathworld "A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions: The empty set is in T. X is in T. The intersection of…
Ahmed Abdullah
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$\mathbb{R}$ closed or open in $\mathbb{R}^2$?

Is $\mathbb{R}$ closed or open in $\mathbb{R}^2$ with respect to the standard topology of $\mathbb{R}^2$ (open sets are the open epsilon-balls)? I think, $\mathbb{R}$ is closed in $\mathbb{R}^2$ because $\mathbb{R}$ can be identified with…
toyboy
  • 71
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Countable Product of Sequentially Compact spaces is Sequentially Compact

I would like to prove that a countable product $$\prod_{i \in \mathbb{N}}X_i $$ of sequentially compact spaces $X_i$ is sequentially compact. That is, any sequence in the product space has a convergent subsequence (where the product has the usual…
mchristos
  • 409
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How to prove that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic?

I'm asked to prove that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic. So far, I've been able to prove that $\mathbb{R}\backslash\{a\}$ and $\mathbb{R}^2\backslash\{b\}$ are not homeomorphic, for $a\in \mathbb{R}$ and $b\in \mathbb{R}^2$. But…
RBS
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Does there exist more than 3 connected open sets in the plane with the same boundary?

I've wondered about the following question, whose answer is perhaps well known (in this case I apologize in advance). The Lakes of Wada are a famous example of three disjoint connected open sets of the plane with the counterintuitive property that…
Malik Younsi
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Every open neighborhood of a point in the closure contains a point in the set

Definitions: Neighborhood: The neighborhood of a point $x$ is a subset of the topological space $X$ that contains an open set such that $x$ is in that open set. Closure: The closure of a set $S$ is the intersection of all the closed subsets of the…
mr eyeglasses
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Stone–Čech remainder of limit ordinals

This is a follow-up question on an earlier question about the Stone–Čech compactification of limit ordinals (Compactifications of limit ordinals): For a limit ordinal $\alpha$, what is the cardinality of its Stone–Čech compactification remainder…
Max
  • 71
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Dense subsets of an infinite set in the cofinite topology

A subset $E \subset X$ of a topological space $X$ is dense if $\overline{E} = X$ where $$ \overline{E} = \bigcap \lbrace C \subseteq X \mid C \text{ is closed and } E \subseteq C \rbrace $$ But in the cofinite topology closed sets are defined to be…
Kevin Johnson
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