Question

Is every disconnected closed set the disjoint union of two nonempty closed sets?

score 1.6k · Answer 1 · 2026-04-09 21:05:39

1.6k

Views

Is every disconnected closed set the disjoint union of two nonempty closed sets?

Published on 09 Apr 2026 - 21:05

score 45 · Answer 2 · 2026-04-12 18:46:30

45

Views

$\exists$ injective map but no bijective map from $A$ to $B$ $\implies |A|<|B|$

Published on 12 Apr 2026 - 18:46

score 348 · Answer 3 · 2026-04-13 18:07:12

348

Views

Every point of discontinuity of a function $f$ is a removable discontinuity. Prove that $g(x)=\lim_{y\to x}f(y)$ is continuous.

Published on 13 Apr 2026 - 18:07

score 285 · Answer 4 · 2026-03-30 15:45:02

285

Views

If $X$ is a non-empty subset of $\mathbf{R}$, show that $X$ is bounded if and only if $\inf(X)$ and $\sup(X)$ are finite.

Published on 30 Mar 2026 - 15:45

score 90 · Answer 5 · 2026-03-26 14:21:06

90

Views

Show that if $X$ is a bounded subset of $\textbf{R}$, then the closure $\overline{X}$ is also bounded.

Published on 26 Mar 2026 - 14:21

score 101 · Answer 6 · 2026-03-30 15:45:02

101

Views

Let $E$ be a bounded subset of $\textbf{R}$, and let $s = \sup(E)$. Show $s$ is an adherent point of $E$, and is also an adherent point of $E^{c}$.

Published on 30 Mar 2026 - 15:45

score 112 · Answer 7 · 2026-04-11 14:50:38

112

Views

Is this another definition for inverse relation?

Published on 11 Apr 2026 - 14:50

score 50 · Answer 8 · 2026-04-10 23:32:12

50

Views

Combinatorics - Inclusion-Exclusion verification

Published on 10 Apr 2026 - 23:32

score 66 · Answer 9 · 2026-03-29 02:36:03

66

Views

Prove. $\frac{n}{n+1}$ is a Cauchy sequence.

Published on 29 Mar 2026 - 2:36

score 66 · Answer 10 · 2026-04-13 19:35:51

66

Views

Could someone help me solve this problem about continuity?

Published on 13 Apr 2026 - 19:35

score 1.3k · Answer 11 · 2026-04-11 01:05:29

1.3k

Views

Proof verification: Composition of maps is associative

Published on 11 Apr 2026 - 1:05

score 44 · Answer 12 · 2020-05-17 03:59:35

44

Views

Calculate the area of a sector of a circle? [My answer is different to the textbook answer, who is correct?]

Published on 17 May 2020 - 3:59

score 599 · Answer 13 · 2026-04-10 04:52:56

599

Views

Proof about Extension and Contraction of Ideals

Published on 10 Apr 2026 - 4:52

score 221 · Answer 14 · 2026-04-10 11:45:36

221

Views

Michael Spivak Calculus Limits problem proof verification

Published on 10 Apr 2026 - 11:45

score 45 · Answer 15 · 2026-04-12 13:31:33

45

Views

Find the convergence interval of the series $ \sum_{n = 0}^{\infty} (\frac {x} {1 + x}) ^n.$

Published on 12 Apr 2026 - 13:31

List Question

Is every disconnected closed set the disjoint union of two nonempty closed sets?

$\exists$ injective map but no bijective map from $A$ to $B$ $\implies |A|<|B|$

Every point of discontinuity of a function $f$ is a removable discontinuity. Prove that $g(x)=\lim_{y\to x}f(y)$ is continuous.

If $X$ is a non-empty subset of $\mathbf{R}$, show that $X$ is bounded if and only if $\inf(X)$ and $\sup(X)$ are finite.

Show that if $X$ is a bounded subset of $\textbf{R}$, then the closure $\overline{X}$ is also bounded.

Let $E$ be a bounded subset of $\textbf{R}$, and let $s = \sup(E)$. Show $s$ is an adherent point of $E$, and is also an adherent point of $E^{c}$.

Is this another definition for inverse relation?

Combinatorics - Inclusion-Exclusion verification

Prove. $\frac{n}{n+1}$ is a Cauchy sequence.

Could someone help me solve this problem about continuity?

Proof verification: Composition of maps is associative

Calculate the area of a sector of a circle? [My answer is different to the textbook answer, who is correct?]

Proof about Extension and Contraction of Ideals

Michael Spivak Calculus Limits problem proof verification

Find the convergence interval of the series $ \sum_{n = 0}^{\infty} (\frac {x} {1 + x}) ^n.$

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