Question

Find $X,Y\subseteq \Bbb{R}$ such that there are $f\colon X\to Y$ and $g\colon Y\to X$, both bijective and continuous but $X\not\cong Y$.

score 368 · Answer 1 · 2022-05-30 04:03:52

368

Views

Find $X,Y\subseteq \Bbb{R}$ such that there are $f\colon X\to Y$ and $g\colon Y\to X$, both bijective and continuous but $X\not\cong Y$.

Published on 30 May 2022 - 4:03

score 90 · Answer 2 · 2026-03-26 21:26:25

90

Views

Show that $\dim(Y)<\infty$, application of the Hahn-Banach theorem.

Published on 26 Mar 2026 - 21:26

score 266 · Answer 3 · 2022-06-06 07:35:08

266

Views

Show that the range $R(T)$ is not closed in $l^2$ norm for $T:(x_1, . . . , x_n, . . .) → (x_1, . . . ,\frac{1}{n}x_n, . . .)$

Published on 06 Jun 2022 - 7:35

score 233 · Answer 4 · 2022-06-07 06:42:44

233

Views

Question in Folland's Proof of Open Mapping Theorem

Published on 07 Jun 2022 - 6:42

score 57 · Answer 5 · 2022-06-11 03:26:30

57

Views

Continuous functions carry compact sets into compact sets. Proof Attempt

Published on 11 Jun 2022 - 3:26

score 57 · Answer 6 · 2022-06-13 01:02:54

57

Views

Are all identification maps on a topology open? (proof verification)

Published on 13 Jun 2022 - 1:02

score 55 · Answer 7 · 2026-03-26 09:40:01

55

Views

Question about continuous but neither open nor closed function on topological space X

Published on 26 Mar 2026 - 9:40

score 36 · Answer 8 · 2022-06-21 23:06:32

36

Views

The property of the saturated sets

Published on 21 Jun 2022 - 23:06

score 23 · Answer 9 · 2022-06-29 21:56:47

23

Views

A homeomorphism essentially takes a topological space into open subspace of the real?

Published on 29 Jun 2022 - 21:56

score 93 · Answer 10 · 2022-07-14 05:21:51

93

Views

$\forall \alpha\in\mathscr{A},\; X_\alpha\cong Y_{\varphi(\alpha)}\implies \prod_\alpha X_\alpha\cong \prod_\beta Y_\beta$ (Homeomorphisms).

Published on 14 Jul 2022 - 5:21

score 112 · Answer 11 · 2022-07-30 16:20:43

112

Views

Determine if analytic mappings from $\mathbb{C}\setminus \{0\}$ and $\mathbb{C}\setminus [0,\infty)$ to open unit disk exist and if so, find them...

Published on 30 Jul 2022 - 16:20

score 49 · Answer 12 · 2022-08-29 06:50:04

49

Views

Bounded surjective linear map from $L^p$ to $L^r$ with $r < p$

Published on 29 Aug 2022 - 6:50

score 43 · Answer 13 · 2022-08-29 20:12:05

43

Views

Example of a function such that the property $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ doesn't hold.

Published on 29 Aug 2022 - 20:12

score 156 · Answer 14 · 2022-09-01 07:17:38

156

Views

How to characterize open sets in the unit circle

Published on 01 Sep 2022 - 7:17

score 36 · Answer 15 · 2026-03-26 09:42:31

36

Views

A question related to the closedness of a map between two topological spaces

Published on 26 Mar 2026 - 9:42

List Question

Find $X,Y\subseteq \Bbb{R}$ such that there are $f\colon X\to Y$ and $g\colon Y\to X$, both bijective and continuous but $X\not\cong Y$.

Show that $\dim(Y)<\infty$, application of the Hahn-Banach theorem.

Show that the range $R(T)$ is not closed in $l^2$ norm for $T:(x_1, . . . , x_n, . . .) → (x_1, . . . ,\frac{1}{n}x_n, . . .)$

Question in Folland's Proof of Open Mapping Theorem

Continuous functions carry compact sets into compact sets. Proof Attempt

Are all identification maps on a topology open? (proof verification)

Question about continuous but neither open nor closed function on topological space X

The property of the saturated sets

A homeomorphism essentially takes a topological space into open subspace of the real?

$\forall \alpha\in\mathscr{A},\; X_\alpha\cong Y_{\varphi(\alpha)}\implies \prod_\alpha X_\alpha\cong \prod_\beta Y_\beta$ (Homeomorphisms).

Determine if analytic mappings from $\mathbb{C}\setminus \{0\}$ and $\mathbb{C}\setminus [0,\infty)$ to open unit disk exist and if so, find them...

Bounded surjective linear map from $L^p$ to $L^r$ with $r < p$

Example of a function such that the property $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ doesn't hold.

How to characterize open sets in the unit circle

A question related to the closedness of a map between two topological spaces

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